Summations in Bernoulli s Triangles via Generating Functions

Transcription

1 Journal of Integer Sequences, Vol 20 (207, Article 73 Summations in Bernoulli s Triangles via Generating Functions Kamilla Oliver 9 Stellenberg Road 730 Somerset West Sout Africa olikamilla@gmailcom Helmut Prodinger Department of Matematical Sciences Stellenbosc University 7602 Stellenbosc Sout Africa proding@sunacza For Pilippe Flajolet (948 20, te master and advocate of generating functions Abstract We revisit sums along straigt lines of indices in Bernoulli triangles, and empasize te use of generating functions as te appropriate tool Tis leads to more direct and extended results, compared wit a recent paper in tis journal

2 Introduction Let ( B [] n n,k k and B [m+] n,k 0 j k n,j, m A recent study [4] refers to tese numbers as te m-t Bernoulli triangle In particular, sums in tem, along certain straigt lines, are studied We revisit tese problems and treat tem wit generating functions because experience sowed as tat suc an approac is well suited for te problems at and It leads to quicker and cleaner proofs and also to extended results We ope tat te readers migt find our treatment interesting As a general reference to te metods tat we use, we would like to mention te book Analytic Combinatorics [] We would also like to give a pointer to te gfun package by Salvy and Zimmermann, as available by now in Maple [5] Our first observation is tat te iterated sums can be reduced to a single summation: B [3] n,k B [4] n,k 0 j k and in general, by induction, Let Ten F (z,y : 0 j k n,k 0 k n B [2] n,k 0 k ( n 0 k (j + 0 k F m (z,y : ( n, (k +, 0 k ( k +m 2 0 k n m 2 n,k zn y k ( n z n y k k 0 n(+y n z n It is even beneficial to introduce a trivariate generating function: H(z,y,w : m F m (z,yw m F (z,y+ m 2 w m 0 k n 0 k ( k +2 2, z(+y ( k +m 2 m 2 z n y k 2

3 Hence z(+y +w m 0 z(+y +w w m 0 k n ( k +m m 0 k n ( n ( w k z n y k z n y k z(+y + w [ ]( n ( w n y n+ y z n y +w 0 n z(+y + w ( w n y n+ (2 w n z n y +w 0 n w (+y n z n y +w z(+y + z(+y w y +w w zy 2zy F m (z,y [w m ] z(+y w y +w 0 n y w w w yz(2 w z(+y w zy 2zy ( zy m z(+y ( 2zy m Wewanttopointoutattatstageowtoextracttediagonal fromabivariategenerating function F(z,y Te procedure was first observed by Hautus and Klarner [3] but is by now almost a standard tool in combinatorial analysis n 0 F(n,nz n 2πi C dt t F( z t,t, were te curve C winds around te origin exactly once Tis integral can be evaluated as a sum over te residues of F( z t,t, over te small solutions t s(z, ie, tose tat satisfy t s(z 0 for z 0 (tese poles stay inside te curve, if it is made sufficiently small We will apply tis principle for trivariate generating functions, were te tird variable w is treated as a parameter 2 A summation Te following sum was of interest for Neiter and Proag [4] n j,k 2j 3

4 We can be careless about te range of summation, since only terms of te form 0 k 2j n j contribute; oters being automatically equal to zero Te extra factor generates tis summation, applied to te generating function zy 2 F m (z,y: ( zy m zy 2 ( 2zy m ( z(+y Since te interest is/was only for n k, we extract te diagonal from tis generating function, or even from te trivariate generating function zy 2 m F m (z,yw m zy 2 z(+y As mentioned before, te diagonal is te contour integral dt 2z 2πi t zt z/t(+t 2z w( z dt 2πi zt w( zy 2zy 2z t z(+t 2z w( z Tere are two poles: t /z and t z/( z Only te second one is small and we compute te residue: { } 2z Res tz/( z ztt z(+t 2z w( z 2z z 2 /( z z 2z w( z 2z z z 2 2z w( z We note for furter reference tat z z 2 m 0 z z 2 n 0 w m ( zm ( 2z m F n+ z n is te generating function of te Fibonacci numbers [2] Let us start wit te simple instance m and extract te coefficient of w m : so tat z z 2, ( n j ( n j F n+, n 2j j 0 j n/2 4

5 wic is classical Now we move to te instance m 2 and extract te coefficient of w m : z z z 2 2z 2 2z +z z z 2 Reading off te coefficient of z n ere leads to wic is te evaluation of 2 n+ F n+ F n 2 n+ F n+2, B [2] n j,n 2j By computing te partial fraction decomposition, te next few instances are (n 2 n +F n+3, (m n n F n+4, (m n n n +F n+5, (m 5 and we wonder if we can say anyting in general We can indeed apply te principle of partial fraction decomposition directly to 2z z z 2 2z w( z w(2 w (+w w 2 ( w z(2 w + zw ( z z 2 (+w w 2 Te second term is easier: [z n w m zw ] ( z z 2 (+w w 2 [wm ] F n+ wf n +w w 2 ( m [w m ] F n+ +wf n w w 2 ( m (F n+ F m +F n F m ( m F n+m Tis is te term tat we observed earlier, from te first few instances Te first term could be expanded to any number of terms, but it does not seem tat a nice closed formula comes out Te sape is polynomial in z of degree m w m ( 2z m m Calling te polynomials in te numerator u m u m (z, Maple s gfun command computes te following recursion: u m+3 zu m+2 (2z (3z 2u m+ (z (2z 2 u m 0, wit initial conditions u 0 0, u 2, u 2 4z 5

6 3 A more general summation Let c be an integer We are, as our predecessors, interested in n cj,k (c+j, especially for k n Since te instance c was discussed at lengt in te previous section, we can be brief ere Te trivariate generating function of interest is now z c y c+ z(+y w( zy 2zy Extracting te diagonal leads to te computation of te residue(s of 2z z c tt z(+t 2z w( z { } 2z Res tz/( z z c tt z(+t 2z w( z 2z z c+ /( z z 2z w( z 2z z z c+ 2z w( z z z c+ m 0 w m ( zm ( 2z m Te first factor generates some generalized Fibonacci number We keep te notation of Neiter and Proag [4]: z c z z c+ n 0 For m, te coefficient of w m leads to z z c+ n 0 λ n (cz n λ n+c (cz n, and tus we ave te first summation B [] n cj,n (c+j λ n+c(c 6

7 For m 2, te coefficient of w m leads to z z z c+ 2z 2c 2 c and te coefficient of z n is given by wic is te summation for 2 c 2 c 2n ( λ 2 c n+c (c+ 2z 2 c c j B [2] n cj,n (c+j We stop ere since furter instances become a bit clumsy 4 Anoter summation + c j 2j z j z z c+ 2 j λ n+c j (c, Te next type of summation tat we will treat wit generating functions is n j,j 0 j n/2 In tis case one as to be careful not to include unwanted terms in te summation Let us start wit m : ( n j ( n j j j 0 j n/2 0 j n We consider (again te generating function ( n j z n ( k z k z j j j 0 j n wence we get F n+ for te summation Now let us move to m 2 B [2] n j,j ( n j 0 j n/2 0 j n 0 j n 0 j n/2 ( n j ( n j j,k 0 n/2<j n 0 n/2<j n j ( n j 2 n j 0 j n z(+z, ( n j 2 (n+/2 + 7

8 Te generating function related to te first term is ( n j z n z ( n+ n z + n 0 0 j n n 0 0 n 0 n z ( k + z k z z + ( z +2 0 k 0 0 ( z 2 z2 z z Tis leads to te evaluation F n +2F n+ 2 (n+/2 + F n+3 2 (n+/2 ( n+ z n + z z z +2 2 z z 2 z In te same style one can also treat te iger instances m 3,4, However, we would like to offer a tool tat is very useful: gfun, implemented in Maple, and in particular its guessing abilities In tis way, one can get results very quickly, altoug formal proofs would still be required Tis leads for m 2 to z +2 z z +2z 2 2z 2, and for m 3 to 3z +5 z z +2z 2 2( 2z 2 7+2z 2 2( 2z 2 2, wit result F n+5 { 2 k (k +8, for n 2k; 2 k (k +7, for n 2k + Te reader can create furter examples erself References [] P Flajolet and R Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009 [2] R L Graam, D E Knut, and O Patasnik, Concrete Matematics, 2nd edition, Addison-Wesley, 994 [3] M L J Hautus and D A Klarner, Te diagonal of a double power series, Duke Mat J 38 (97, [4] D Neiter and A Proag, Links between sums over pats in Bernoulli s triangles and Fibonacci numbers, J Integer Sequences 9 (206, Article 683 8

9 [5] Bruno Salvy and Paul Zimmermann, Gfun: a Maple package for te manipulation of generating and olonomic functions in one variable, ACM Trans Mat Software 20 (994, Matematics Subject Classification: Primary 05A5; Secondary 05A9, B39 Keywords: Bernoulli triangle, generating function, diagonal of a double power series (Concerned wit sequence A00008 Received November Revised version received December Publised in Journal of Integer Sequences, December Return to Journal of Integer Sequences ome page 9