INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV 26506, USA gould@math.wvu.edu Jocely Quaitace Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV 26506, USA jquaita@math.wvu.edu Received: 9/1/07, Revised: 11/12/07, Accepted: 12/6/07, Published: 12/20/07 Abstract Defie {}, the floor sequece, by the liear recurrece f( + 1) = f(), f(1) = 1. Similarly, defie {g()}, the roof sequece, by the liear recurrece g( + 1) = g(), g(1) = 1. This paper studies various properties of these two sequeces, icludig prime criteria, asymptotic approximatios of { f(+1) } ad { g(+1) g(), ad the iteratio coefficiets associated with f( + r) ad g( + r), for ay r 1. } 1. Itroductio The Bell umbers may be defied by the liear recurrece B( + 1) = ( =0 ) B(), (1.1) with the iitial coditio that B(0) = 1. These umbers, 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147,... have bee extesively studied i [2]. They may also be defied by the expoetial geeratig
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 2 fuctio =0 B() t! = exp(et 1). (1.2) I [7], we studied the followig geeralizatio of Equatio (1.1). Let {h()} =0 be the sequece defied by the liear differece equatio ( ) h( + 1) = h(), (1.3) =0 where h(0) = a ad h(1) = b. Note, if h(0) = 1 ad h(1) = 1, Equatio (1.3) becomes (1.1). Through successive iteratios of (1.3), we were able to show that h( + r) = r 1 ( + j h() A r j() j=0 where the A r j() are polyomials i that satisfy A r+1 j () = r j 1 i=0 ( + r i ), r 1, 1, (1.4) ) A r i j (). (1.5) Oe reaso the A r j() are importat is that they provide a ew partitio of the Bell umbers which is remiiscet of the formula B() = S(, ). (1.6) I Equatio (1.6), S(, ) is the appropriate Stirlig umber of the secod id. I particular, we have show that [7] =0 1 B() = A j (0) (1.7) j=0 B() = A +1 0 ( 1). (1.8) ( ) Whe aalyzig the proof of Equatio (1.4), we realized that the i (1.3) ca be ( ) replaced by A(, ), where A(, ) is a arbitrary fuctio of [3]. Because is closely related to by the relatio [1] ( ) we thought it would be atural to let A(, ) =. mod, (1.9)
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 3 Thus, we decided to study two particular fuctios. aalog of (1.1), is defied by f( + 1) = f() = The first fuctio, a floor fuctio f(d), (1.10) with the iitial coditio of f(1) = 1. The so geerated, 1, 1, 3, 7, 16, 33, 71, 143, 295, 594, 1206, 2413, 4871, 9743, 19559,..., evidetly have ot bee studied i the literature ad were ot foud i the Olie Ecyclopedia of Iteger Sequeces (OEIS). We will call the {} the floor sequece. We also defie a roof fuctio aalog of (1.1), amely j=1 d j g( + 1) = g(), (1.11) with the iitial coditio that g(1) = 1. Recall that deotes the least iteger greater tha or equal to. The g() so geerated by 1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157,... behave somewhat differetly tha the. They also were ot foud i the OEIS. We call {g()} the roof sequece. Because x = x, there are iterestig relatioships betwee floor ad roof. Note that give positive itegers ad, it is easy to verify that + 1 1 = = + 1. (1.12) Equatio (1.12) allows us to form a alterative recurrece formula for the roof sequece, amely g( + 1) = 1 1 g() + g(), 2. (1.13) If we adopt the covetio that the secod sum o the right is vacuous whe = 1, Relatio (1.13) is true for 1. This paper has four mai sectios. I Sectio 2, we prove prime criteria for the floor sequece ad the roof sequece. These criteria are remiiscet of the prime criteria discussed i [4] ad [5]. I Sectio 3, we discuss the asymptotic ature of ad g(). I Sectio 4, we aalyze the ordiary geeratig fuctios associated with ad g(). Fially, i Sectio 5, we give formulas that relate f( + r) ad g( + r) bac to ad g(). These iteratio formulas are similar to Equatios (1.3) ad (1.4).
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 4 2. Prime Number Criteria for the Floor ad Roof Sequeces For the floor sequece defied by (1.10), we state a useful prime umber criterio. The proof of this criterio uses the followig lemma. Lemma 2.1 Let be as defied by (1.10). The, for 2, f( + 1) = d f(d) (2.1) Proof of Lemma 2.1. We have f( + 1) = = 1 f() ( 1 = d f(d). 1 ) f() f() The last equality follows because 1 = { 1, if ; 0, if. Remar 2.1 Lemma 2.1 is a alterative recurrece that shows how to calculate f( + 1) from. Theorem 2.1 (Prime umber criterio for the floor sequece) Let be the fuctio defied by Equatio (1.10). The is prime if ad oly if f( + 1) = 2 + 1. (2.2) Proof of Theorem 2.1. Whe is prime, the oly divisors of are itself ad 1. Thus, Equatio (2.1) implies f( + 1) = d f(d) = f(1) +, which is simply Equatio (2.2) restated. O the other had, if is ot prime the at least oe of the positive summads o the right side of Equatio (2.1), other tha d = 1 ad d =, is o-zero. This meas that Equatio (2.2) could ot hold.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 5 Remar 2.2 Relatios (2.1) ad (2.2) show that the sequece defied by icreases somewhat faster tha 2 1 for 6. We ow tur to the roof sequece, amely g() defied by (1.11), ad fid the correspodig versios of Lemma 2.1 ad Theorem 2.2. Lemma 2.2 Let g() be as defied by (1.11). The, for 2, g( + 1) = 2g() + g(d) (2.3) d ( 1) Proof of Lemma 2.2. We have g( + 1) g() = The last equality follows because 1 g() 1 g() 1 ( ) 1 = g() + g() = g() + g(d) 1 = d ( 1) { 1, if ( 1); 0, if ( 1). We o loger detect a simple prime criterio for the roof sequece. We shall be cotet with just the followig theorem, whose proof follows directly from Lemma 2.2. Theorem 2.2 (Prime umber criterio for roof sequece) Let g() be the fuctio defied by Equatio (1.11). The 1 is prime if ad oly if g( + 1) = 2g() + g( 1) + 1. (2.4) Remar 2.3 We see from Relatios (2.3) ad (2.4) that g() icreases cosiderably faster tha 2 1 for 5.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 6 3. Growth Estimates for ad g() Usig a Third Sequece We defie a fractio aalog h() of (1.10) ad (1.11) by h( + 1) = h(), (3.1) with the iitial coditio that h(1) = 1. Sice we are worig with all positive umbers, the, (3.2) The first few values of the h sequece are 1, 1, 3, 7.5, 17.5, 39.375, 86.625, 187.8675, ad 402.1875. Theorem 3.1 (Recurrece relatio for h()) For all 2, Proof of Theorem 3.1. By (3.1), we have h( + 1) = 2 1 h(). (3.3) 1 h( + 1) = 1 h() = h() + h() = h() + 1 1 1 h() = h() + 1 h() = 2 1 1 h(), which is precisely (3.3). From (3.3), we have immediately obtai h(+1) h() = 2 1, givig us the followig result. 1 { } Theorem 3.2 (Limit of h(+1) ) The sequece h(+1) h() h() =3 approaches 2 as. is a decreasig sequece that Applyig (3.3) iteratively leads to the explicit formula give i the ext theorem. Theorem 3.3 (Explicit formula for h()) For all 0, h( + 2) = (2+2)!!(+1)!2 +1.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 7 Remar 3.1 Substitutig (3.4) bac ito (3.1) gives the biomial idetity + =2 (2 2)! ( 2)!( 1)!2 = (2)! 1 ( 1)!!2 for 2. This may be restated i biomial coefficiet form as =2 ( 2 2 ) ( 2 2 +1 = ) 2 (3.4) for 2, ad does ot appear i [8]. The idetity may be proved quicly by iductio. It is a routie calculatio with the biomial series to use (3.4) to establish the followig geeratig fuctio result. Theorem 3.4 (Geeratig fuctio for h()) h( + 1)x = x. (3.5) (1 2x) 3 2 Remar 3.2 I aalogy to (3.1), (3.2), ad (3.3), it is ot difficult to use the biomial expasio (4.1) to obtai a geeratig fuctio for. = x = 1 1 x + x (1 x). (3.6) 2 From these results, ad umerical tables, we are led to the followig result. Theorem 3.5 (Bouds for ratios of successive terms) For all 4, the sequeces f, h, ad g satisfy Moreover, for all for 4, we have < h() < g() (3.7) 2 < f( + 1) < h( + 1) h() < g( + 1). (3.8) g() Furthermore, lim f(+1) = 2 ad lim h(+1) h() = 2. Table 2 exhibits values of the ratios i Equatio (3.8) for = 1, 2,..., 24.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 8 The followig 4 lemmas will prove Theorem 3.5. Lemma 3.1 Let, g(), ad h() be as previously defied. The, for 4, < h() < g(). Proof of Lemma 3.1. The proof will use mathematical iductio. For example, i order to show that < h(), we ote that f(4) = 7 < 7.5 = h(4). We ow assume the iductio hypothesis, i.e., for all iteger values greater tha 4 ad less the or equal to, < h(). Thus, f( + 1) = 1 f() = f(1) + f() + f(1) + 1 =2 1 < h(1) + =2 1 < h(1) + =2 =2 f() + f() + h() h() + h() = h( + 1). The iequality betwee the first ad secod lies comes from (3.2), while the other two iequalities are a result of the iductio hypothesis. The proof of h() < g() is similar ad will be omitted. Lemma 3.2 For 4, we have 2 < f(+1). Proof of Lemma 3.2. We wat to show f(+1) > 2 2 1 = 2 1 1 ; that is, we wat to show 1 1 f( + 1) 2 1 > 1 1. (3.9) However, the calculatios of Theorem 2.1 imply, for 4, that f( + 1) 2 > 0. Thus, f( + 1) 2 = f( + 1) + 2 + 2 1 2 + 1 = f( + 1) + + 1 1 > 0. The right had iequality is simply a restatemet of (3.10), which proves our claim. Lemma 3.3 For 4, we have f(+1) < h(+1) h().
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 9 Proof of Lemma 3.3. Table 2 shows, for 4 < 15, that f(+1) 15. By Equatio (3.3), it is sufficiet to show f(+1) show 2 1 1 < h(+1). So we ow assume h() < 0, i.e., it is sufficiet to f( + 1) 2 < 1. (3.10) By Equatio (2.1), we ow f( + 1) 2 = 1 + d f(d). Thus, (3.11) becomes d 1, Sice the largest possible divisor i 1 + d d 1, We claim that ( 1) 1 + f(d) < = 1 1 d d 1, f(d) = d d [ 1 + ] d f(d) is 2, we have d 1, ( ) ( ) 2 f(d) f 1 f 1 = 2 2 d d i=1 f(). (3.11) ( ) f. 2 2 ( ) ( 1) f < f( 1), 15 (3.12) 2 2 The justificatio for (3.12) is as follows. First, rewrite Equatio (3.12) as By Remar 2.2, we ow ( 1) < 2 f( 1) f ( ). (3.13) 2 2 1 2 f( 1) < f ( ), 12. (3.14) 2 The by a simple iductio argumet, it is easy to show, for 15, that ( 1) < 2 1 2 f( 1) < 2 f ( ). 2 By combiig the previous calculatios, we have, for 15, ( 1) 1 + f(d) ( ) ( 1) 1 f < f( 1) 2 2 d d 1, which is Equatio (3.12). 1 =,
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 10 Remar 3.3 I the proof of Lemma 3.3, we showed that {} is a icreasig sequece. Remar 3.4 Usig Theorem 3.2, Lemma 3.2, Lemma 3.3, ad The Squeeze Theorem, we teds to the limit 2 as icreases idefiitely. f(+1) Lemma 3.4 For 4, we have h( + 1) h() < g( + 1). g() Proof of Lemma 3.4. The result is clearly true whe = 4. We ow assume > 4. Usig Theorem 3.2, it suffices to show that g(+1) > 2 1 i.e., it suffices to show g() 1 By (2.4), we ca rewrite (3.15) as g( + 1) g() > g() 1. (3.15) ( 1) d ( 1) g(d) > g(). (3.16) Clearly, ( 1) d ( 1) g(d) ( 1)g( 1). (3.17) Thus, we ow wat to show ( 1)g( 1) > g(). (3.18) I order to prove (3.18), ad thus fiish the proof of Lemma 3.4, we first ote that (3.18) is equivalet to i.e., 2 1 g( 1) g( 1) > g( 1) + g(), Thus, provig (3.18) is equivalet to provig 2 1 g( 1) > 2g( 1) + g(). 2 1 2 2 g() < ( 2) g(). (3.19)
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 11 A term-by-term compariso of the sums i (3.19) implies that we must show 1 2 < ( 2). (3.20) Note that 1 = 2 + 1. Usig properties of the floor ad the fact that x = x, it is easy to show that for all real umbers a ad b, a+b a + b. Thus, the previous lie implies, sice 1 2, 1 2 2 2 2 1 + + < ( 2), which is a restatemet of (3.20). 3.1. Ope Questios By aalyzig the ratios i Table 2, we form the followig cojectures, whose proofs remai ope questios. Cojecture 3.1 The sequece Cojecture 3.2 The sequece of 1 + 2. A plausibility argumet for the limit g(+1) g() = p beig a prime, we fid { } f(+1) alterately icreases the decreases to its limit. =4 { } g(+1) alterately icreases the decreases to the limit g() =4 g(p + 2) g(p + 1) = 2 + 1 g(p+1) g(p) may ru as follows. From Equatio (2.4), with It is easy to show that g(p) is a icreasig sequece. Thus, if g(+1) g() to the equatio L = 2 + 1, from which we deduce that L = 1 + 2. L Oe possible way to compute the limit of sequece for g(). I particular g( + 1) = 1 < 1 g() = 1 { g(+1) g() + } 1 g() + g() = =4 1 g(p + 1). (3.21) has limit L, we are led is to use (1.13) to obtai a boudig g() + 1 g() 1 + g() + g().
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 12 We may the defie a boudig sequece M() as follows: M( + 1) = 1 1 + M() + M(), (3.22) with iitial coditio that M(1) = 1. The, g() < M() for all 5. This sequece has the values 1, 1, 3, 8, 20.5, 51.5, 128, 316.1,... Lemma 3.5 (Limit of M(+1) ) If M() ad lim 1 M( + 2) M() = 0 (3.23) M( + 1) lim = L (3.24) M() exist, the L = 1 + 2. Proof of Lemma 3.5. From (3.23), we have M( + 1) = 1 2 M() = Subtractig (3.25) from (3.26), we fid which the gives M( + 1) M() = This the yields the relatio 1 1 1 2 M() + M(), (3.25) 1 M() + M(). (3.26) 2 M() M( + 1) = 2M() + M( 1) + 2 M() + M(), 2 M(). M( + 1) M() = 2 + 1 M() M( 1) + 1 2 M() M(). (3.27) Therefore, if we assume the limit (3.24) exists ad that M(+1) M() that L = 2 + 1/L, which gives L = 1 + 2. approaches a limit L, we fid
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 13 Remar 3.5 (The coverse of Lemma 3.5) We fid from (3.28) that if M( + 1)/M() approaches a limit L, ad if the limit i (3.24) is R, the R = L 2 1/L, from which we could compute L if we ew R. Thus, from Lemma 3.5, we see that the M() sequece would be useful if we could show that 1 + 2 < g( + 1) g() < M( + 1). M() The, by the Squeeze Theorem, we would a have a proof of Cojecture 2. 3.2. Asymptotic Tables Table 1 below gives values for the four sequeces. Table 2 gives values of the ratios i (3.9) ad M(+1) M(). f h g M() 1 1 1 1 1 2 1 1 1 1 3 3 3 3 3 4 7 7.5 8 8 5 16 17.5 20 20.5 6 33 39.375 50 51.5 7 71 86.625 121 128 8 143 187.6875 297 316.1 9 295 402.1875 716 777.3833333 10 594 854.6484375 1739 1906.335714 11 1206 1804.257812 4198 4665.036310 12 2413 3788.941406 10157 11397.76581 13 4871 7922.332031 24513 27812.55897 14 9743 16504.85840 59246 67798.969 15 19559 34279.32129 143006 1.651363960 10 5 16 39138 71007.16553 345381 4.019370878 10 5 17 78428 1.467481421 10 5 833792 9.777186817 10 5 18 156857 3.026680431 10 5 2013272 2.377091654 10 6 19 314047 6.231400887 10 5 4860337 5.776740262 10 6 20 628095 1.280899071 10 6 11734717 1.403292331 10 7 21 1256809 2.629213883 10 6 28329772 3.407699867 10 7 22 2513693 5.389888460 10 6 68396030 8.272537140 10 7 23 5028594 1.103643827 10 7 165121957 2.007678384 10 8 24 10057189 2.257453283 10 7 398644144 4.871238593 10 8 Table 1: The Sequeces, h(), g(), ad M() for = 1, 2, 3,...24
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 14 f(+1) h(+1) h() g(+1) g() M(+1) M() 1 1.000000000 1.000000000 1.000000000 1.000000000 2 1.000000000 1.000000000 1.000000000 1.000000000 3 3.000000000 3.000000000 3.000000000 3.000000000 4 2.333333333 2.500000000 2.666666666 2.666666667 5 2.285714286 2.333333333 2.500000000 2.562500000 6 2.062500000 2.250000000 2.500000000 2.512195122 7 2.151515152 2.200000000 2.420000000 2.485436893 8 2.014084507 2.166666666 2.454545455 2.469531250 9 2.062937063 2.142857143 2.410774411 2.459295582 10 2.013559322 2.125000000 2.428770950 2.452246701 11 2.030303030 2.111111111 2.414031052 2.447122128 12 2.000829187 2.100000000 2.419485469 2.443231960 13 2.018648985 2.090909091 2.413409471 2.440176385 14 2.000205297 2.083333333 2.416921633 2.437710571 15 2.007492559 2.076923077 2.413766330 2.435677098 16 2.001022547 2.071428571 2.415150413 2.433970326 17 2.003883694 2.066666667 2.414122375 2.432516708 18 2.000012751 2.062500000 2.414597406 2.431263408 19 2.002122953 2.058823529 2.414148212 2.430171445 20 2.000003184 2.055555556 2.414383406 2.429211402 21 2.000985520 2.052631579 2.414184509 2.428360642 22 2.000059675 2.050000000 2.414280990 2.427601450 23 2.000480568 2.047619048 2.414203821 2.426919759 24 2.000000199 2.045454545 2.414240669 2.426304249 25 2.000258621 2.043478261 2.414208914 2.425745711 26 2.000000845 2.041666667 2.414225288 2.425236570 Table 2: The Ratios f(+1), h(+1) h(), g(+1) g(), M(+1) M() for = 1, 2, 3,...26 4. Geeratig Fuctios for the Floor ad Roof Sequeces Recall that a variatio of the biomial theorem ([1]) is ( ) x 1 =. (4.1) (1 x) +1 = The floor fuctio aalog of Equatio (4.1) ([1]) is = x = 1 (1 x)(1 x ), (4.2)
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 15 while the roof fuctio aalog of (4.1) is = x = 1 + x x (1 x)(1 x ). (4.3) Remar 4.1 The idea of studyig the floor sum i Equatio (1.10) is ot etirely ew. Relatio (4.2) was used i [6] to study the series trasform G() = F (), 1, (4.4) which has the iverse F () = d ( ) µ [G(d) G(d 1)], 1. (4.5) d What is ovel ow is whe, i Equatio (4.4), we mae G() = F (), Equatio (4.5) does ot hold. For the defied by (1.10), we defie the ordiary geeratig fuctio Usig (4.2), we fid t f( + 1) = t = 1 = 1 F (t) = t. (4.6) f() = f()t = t f()t = 1 f() + 1 f()(t r ) = 1 F (t r ). r=1 r=1 f() O the other had t f( + 1) = =2 t 1 = 1 t t 1 = 1 F (t) 1, t so that the geeratig fuctio for the floor sequece must satisfy the fuctioal equatio 1 t F (t) 1 = 1 F (t r ). (4.7) r=1
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 16 For the g() defied by (1.11), we defie the ordiary geeratig fuctio Equatio (4.3) implies t g( + 1) = O the other had, t = 1 = 1 G(t) = g()t. (4.8) g() = g()t = t + t + 2 g() g()t t g() + g()(t r ) = G(t) + t = G(t) + t r=1 G(t r ). r=1 t t g() t g( + 1) = =2 t 1 g() = 1 t t g() 1 = 1 G(t) 1. t Thus, the geeratig fuctio for the roof sequece must satisfy 1 2t t() G(t) 1 = t G(t r ). (4.9) r=1 5. Expasios Ivolvig f( + r) ad g( + r) I [3], we studied a class of fuctios, H, defied by the relatioship H( + 1) = A(, )H(), (5.1) =0 where A(, ) is a arbitrary fuctio of. I this situatio, we ca let A(, ) =, H(0) = 0, ad H(1) = 1 or A(, ) =, H(0) = 0, ad H(1) = 1. It is a easy exercise to restate Theorems 2.1 to 2.4 of [3] i the cotext of the floor fuctio ad the roof fuctio. These restatemets are recorded below as Corollaries 5.1 to 5.5 respectively.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 17 Corollary 5.1 Let f be the fuctio defied by Equatio (1.10). Let g be the fuctio defied by Equatio (1.11). Let r be a positive iteger. There exist fuctios of, amely A r j() ad Cj r (), such that, f( + r) = g( + r) = r 1 + j f() A r j() j=0 r 1 + j g() Cj r () j=0 where A r j() ad C r j () satisfy the recurrece relatios r j 1 A r+1 j () = A r i j () i=0 r j 1 C r+1 j () = i=0 C r i j () + r + r i + r + r i, r 1, 1, (5.2), r 1, 1, (5.3), 0 j r 1, (5.4), 0 j r 1. (5.5) Note that A r r 1() = 1 ad A r j() = 0 if j < 0 or j > r 1. A similar statemet holds for the C r j (). Corollary 5.2 (The Shift Theorem of [3]) The A r j() (ad C r j ()) coefficiets satisfy the relatio A r+1 j+1 () = Ar j( + 1), j 1, r 0. (5.6) Corollary 5.3 r 1 j A r 1(0) = A r j(0), 0 < < r 1 (5.7) j= r 1 j C 1(0) r = Cj r (0), 0 < < r 1. (5.8) j= Corollary 5.4 (The Iversio Theorem for the Floor Fuctio) We have r 1 F (r) = A r j(0)g(j), r 1 (5.9) j=0 if ad oly if G(r) = F (r + 1) r j=1 r F (j), with G(0) = F (1). (5.10) j
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 18 Corollary 5.5 (The Iversio Theorem for the Roof Fuctio) We have r 1 F (r) = Cj r (0)G(j), r 1 (5.11) j=0 if ad oly if G(r) = F (r + 1) r j=1 r F (j), with G(0) = F (1). (5.12) j Below we provide a table of the A r j(0) ad a table of the C r j (0). Usig Equatios (5.4) ad (5.5) it is easy to show that = A 0(0) ad g() = C 0 (0). I other words, ote that the left most diagoal of the table is our sequece or g(). 1 1 1 3 1 1 7 2 1 1 16 5 2 1 1 33 10 4 2 1 1 71 22 9 4 2 1 1 143 44 18 8 4 2 1 1 Table 3: Values of A r j (0). 1 1 1 3 1 1 8 3 1 1 20 7 3 1 1 50 18 7 3 1 1 121 43 17 7 3 1 1 297 106 42 17 7 3 1 1 Table 4: Values of C r j (0) For Tables 3 ad 4, rows correspod to r = 1, 2, 3,... ad diagoals to j = 0, 1,..., r 1. Ispectio of Tables 3 ad 4 suggests that as, the rows ted to stabilize to a fixed sequece. I the case of Table 3, the fixed sequece is b = 2. Thus, we have the followig theorem.
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 19 Theorem 5.1 Let 0. The, A 2+2 (0) = 2. For Table 4, the stabilizatio sequece is more complicated. followig theorem. I particular, we have the Theorem 5.2 Let a be the sequece defied by the recursio a = 2a 1 + a 2, a 0 = 1, a 1 = 1. The, for 2, C 2 1 2 (0) = a. We will prove Theorem 5.2. Sice the proof of Theorem 5.1 is similar, its proof is omitted. I order to prove Theorem 5.2, we will eed the followig two lemmas. Lemma 5.1 Let 2. Let i 0 ad 0. The, 2 + i 2 = 2 + i 2 (5.13) Proof of Lemma 5.1. Clearly Equatio (5.13) is true whe = 0. Now we assume 1, or equivaletly, 1. By addig 2 to each term of the iequality, we obtai 2 > 2 1 2. I other words, 1 2 < 1 2 1 1 2 1. Multiply each term i the above iequality by 2 to obtai 1 < 2 2 1 2 2 Thus, Equatio (5.14) implies that if 1, 2 = 2. 2. (5.14) We ow wat to obtai the left side of (5.13). Oce agai, assume 1, or equivaletly. 1. Add 2 +i to each term ad obtai 2 +i > 2 +i 1 2 +i +i. 1 Thus, < 1 1 1. Multiply each term i the above iequality by 2 + i 2+i 2+i 1 2+i +i to obtai 1 < 2 + i 2 + i 1 2 + i 2 + i 2 + i + i = 2 i + i 2. (5.15) Thus, Equatio (5.15) implies that if 1, 2+i = 2. Combiig (5.14) ad (5.15) proves the lemma. Lemma 5.2 (Stabilizatio of the left to right diagoal) Let 2 ad i 0. The, C 2 1 2 (0) = C 2 1+i (0). (5.16) 2+i
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 20 Proof of Lemma 5.2. We use iductio o. If = 2, it easy to show, via Equatio (5.5), that C0(0) 3 = 3 = C 3+i i (0). We ow assume (5.16) is true for all iteger values less tha or equal to. I otherwords, we assume the first left to right diagoals of Table 4 stabilize to a fixed umber. By Equatio (5.5), 2 + i C 2+1+i 1+i (0) = C 2+i 1+i (0) 2 + i =0 2 + i = C 1 2 (0), iductive hypothesis 2 + i =0 2 = C 1 2 (0), by, Lemma 4.1 2 =0 = C 2+1 1 (0) by (5.5). The above calculatios prove the lemma. Remar 5.1 Lemma 5.2 implies that for 4, ad i 0, C 2 5 4 (0) = C 2 5+i 4+i (0). Proof of Theorem 5.2. Note that C 3 0(0) = 3 = a 2 ad C 5 1(0) = 7 = a 3. If we ca show, for 4, that C 2 1 2 (0) = 2C 2 3 3 (0) + C 2 5 4 (0), (5.17) we will prove the theorem, sice both the C 2 2 1 (0) ad a obey the same recursio relatio ad have the same iitial coditios. By Equatio (5.8), C 2 1 2 (0) = 2 2 j= 1 C 2 1 j (0) j= 2 j 1 = C 2 1 1 (0) + 2C 2 1 (0) + 2 2 2 j=+1 The last equality is a simple adaptatio of Lemma 5.1. Also, by (5.8), we have 2 4 j C 3 2 3 (0) = C 2 3 j (0) 2 = C 2 3 2 (0) + 2 = C 2 1 (0) + 2 2 4 j= 1 2 4 j= 1 C 2 3 j (0) C 2 1 j+2 (0) = C2 1 (0) + 2 2 2 j=+1 C 2 1 j (0). (5.18) C 2 1 j (0). The third equality comes from lettig i = 2 i Remar 5.1. Thus, the precedig calculatios imply that (5.18) is, i fact, C 2 2 1 (0) = C 1 2 1 (0) + C 2 1 (0) + C 3 2 3 (0). By repeated applicatios of Lemma 5.2, the above equatio becomes C 2 2 1 (0) = C 3 2 3 (0) + C 2 1 (0) + C 3 2 3 (0) = 2C 3 2 3 (0) + C 4 2 5 (0). which is exactly Equatio (5.17)..
INTEGERS:ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 21 Remar 5.2 Lemma 5.2 implies, that for all i 0, a = C 2 2 1 (0) = C 2 1+i 2+i (0). Thus, the proof of Theorem 5.2 shows that a = C 2 1+i 2+i (0), wheever i 0 ad 2. Similiarly, we ca show that wheever i 0, A 2+2+i +i (0) = 2. We ed this sectio with Corollary 5.6, which is a result of Theorems 5.1 ad 5.2. This corollary states that the limitig sequece is the cetral colum of Tables 3 ad 4. Corollary 5.6 Let 0. Let a ad b be as defied i Theorems 5.1 ad 5.2. The, b = A 2+1 (0) ad a = C 2+1 (0). 6. Closig Remars: A Combiatorial Questio There remais the problem of determiig combiatorial structures eumerated by ad g(). Although h() is fractioal, removig the powers of 2 i (3.4) ad relabelig, the sequece a() = ( ) 2, which has the values 2, 12, 60, 280, 1260, 5544, 24024, 102960,... does have some iterest. This sequece is A005430 i the OEIS ad the umbers are sometimes called Apery umbers sice they occur i Roger Apery s proof of the irratioality of ζ(3). Refereces 1. H. W. Gould, Biomial Coefficiets, the Bracet Fuctio, ad Compositios with Relatively Prime Summads, Fiboacci Quarterly, 2 (1964), 241-260 2. H. W. Gould, Bell ad Catala Numbers - Research Bibliography of Two Special Number Sequeces, Fifth Editio, 22 April 1985. x + 43pp. Published by the author, Morgatow, W. Va. 3. H. W. Gould ad Jocely Quaitace, The Geeralized Bell Number Recurrece ad The Geeralized Catala Recurrece, Applicable Aalysis ad Discrete Mathematics, to appear 4. Temba Shohiwa, Ivestigatios i Number Theoretic Fuctios, Ph.D. Dissertatio, West Virgiia Uiversity 1996 5. Temba Shohiwa, O a Class of Prime-Detectig Cogrueces, Discrete Math, 204 (1999), 357-368 6. H. W. Gould, A Bracet Fuctio Trasform ad Its Iverse, Fiboacci Quarterly, 32 (1994), 176-179 7. H. W. Gould ad Jocely Quaitace, A Liear Biomial Recurrece ad the Bell Numbers ad Polyomials, Applicable Aalysis ad Discrete Mathematics, 1, (2007), No. 2. Available at http://pefmath.etf.bg.ac.yu. 8. H. W. Gould, Combiatorial Idetities, A Stadardized Set of Tables Listig 500 Biomial Coefficiet Summatios, Secod Editio, 1972, viii + 106 pp., Published by the author, Morgatow, W.Va.