GENERATING FUNCTIONS OF CENTRAL VALUES IN GENERALIZED PASCAL TRIANGLES. CLAUDIA SMITH and VERNER E. HOGGATT, JR.

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1 GENERATING FUNCTIONS OF CENTRAL VALUES CLAUDIA SMITH and VERNER E. HOGGATT, JR. San Jose State University, San Jose, CA 95. INTRODUCTION In this paper we shall examine the generating functions of the central (maximal) values in Pascal s binomial and trinomial triangles. We shall compare the generating functions to the generating functions obtained from partition sums in Pascal f s triangles. Generalized Pascal triangles arise from the multinomial coefficients obtained by the expansion of ( + x + x + + x*- ), j >_, n >_ 0, where "n" denotes the row in each triangle. For j =, the binomial coefficients give rise to the following triangle: 4 4 etc. For j =, the trinomial coefficients produce the following triangle: 7 The partition sums are defined where M rr i\ S(n,j,k,r) = [ r + n i / c I ; 0 < r < k -, M = [ U ~ ^ ~ r ], [ ] denoting the greatest integer function. To clarify, we give a numerical example. Consider 5(,,4,). This denotes the partition sums in the sixth row of the trinomial triangle in which every fourth element is added, beginning with the second column. The 5(,,4,) = =. (Conventionally, the column of l f s at the far left is the 0th column and the top row is the 0th row.) In the nth row of the j-nomial triangle the sum of the elements is j n. This is expressed by Let S(n,j,k,0) +S(n 9 j,k 9 l) + + S(n,j,k 9 k -) = j". S(nj 9 k 9 0) = (j n + A n )/k 58

2 Feb. 979 GENERATING FUNCTIONS OF CENTRAL VALUES 59 Since S(Q,j 9 k,q) =, S(nJ,k,l) = U n + B n )/k... S(nJ,k,k-l) = (j n + Z n )/k. s(o,j,k,i) = 0... s(o 9 j 9 k 9 k-i) = 0, w e can solve for A Q 9 B Q9..., Z 0 to get 4 0 = k -, Z >o = "!>..., z 0 = -. N o w a departure table can be f o r m e d w i t h A Q 9 B 0,..,.. Z 0 as the 0th row. The term "departure" refers to the q u a n t i t i e s, A B..., Z n that d e - part from the average v a l u e j n /k. Pascal's rule of addition is the simplest method for finding the s u c c e s s i v e rows in each departure table. The d e p a r - ture tables for 5 and 0 p a r t i t i o n s in the b i n o m i a l t r i a n g l e appear b e l o w. Notice the appearance of Fibonacci and Lucas n u m b e r s. Table SUMS OF FIVE PARTITIONS IN THE BINOMIAL TRIANGLE Table SUMS OF TEN PARTITIONS IN THE BINOMIAL TRIANGLE The primary purpose of this paper is to show that the limit of the g e n - erating f u n c t i o n s for the (H - L) / k s e q u e n c e s is precisely the generating functions for the c e n t r a l values in the rows of the binomial and trinomial triangles. T h e (H - L)Ik sequences are obtained from the difference of the m a x i m u m and m i n i m u m v a l u e sequences in a departure table, divided by k 9 w h e r e k denotes the number of p a r t i t i o n s.. GENERATING FUNCTIONS OF THE (H - L)Ik SEQUENCES IN THE BINOMIAL TRIANGLE Table is a table of the (H - L)Ik sequences for k = to k = 5 p a r - titions. The generating function of the m a x i m u m values in the b i n o m i a l triangle is + x - A - kx 7 A - 4x V ^ W e shall examine this and show it to b e the limit of the generating functions of the (H - L)Ik s e q u e n c e s.

3 0 GENERATING FUNCTIONS OF CENTRAL VALUES [Feb. Table (H - L)/k SEQUENCES FOR k = TO k = * Consider the relation S n + = S n + - x S n, The two roots are Therefore, K - K + x = 0. expressed by the equation v i + A - kx i - A - kx X i = a nd K =, K x > K. S n+ lim Q = K } = L by Gauss's theorem that the limit is the root of the maximum modulus. The generating functions for the odd partitions, k = m +, were found to have the form bm >m - XOrr, The generating functions for the even partitions, k = m 9 were found to have the form S m - + Sm - S m x S J m We show these two forms have the same limit. S n - ^n- lim tt-*-oo >n X&n l >n - S n.

4 979] GENERATING FUNCTIONS OF CENTRAL VALUES S n _ ± xs n _ where >n-l L - x + $n X S n _ i + / T i S.n - $n - S n % ^n - ^ n - ^rc - 4a: - x (l + x - A - L + a: r r L - x L - x - x + / l - 4a: 4a: \ We pause now to consider the generating function for + x + x z + 0a: d + at ( )x n + > / l - 4a; (see [ ], p. 4 ). Now the Catalan number ( generating function is n + l \ n / & Thus, ^/ \ - / l - 4a: C ( X ) = ^ / l - 4 ] r r ) = + x + 0a: + 5a: + x (see [], p. 8 ). We observe the following relationship between these two series: ( + x + x + 0a: + 7 (to * + - l)/a? a: ( + a: + 0x + 5a: + a: Next we wish to blend these two series. Replace x with x /l - /l - 4a: \ /I - 4a: /l - 4a: + a: + a: 4 + 0a: + a: 8 + We multiply the latter by x y after replacing x with x. X kx X A - 4a: V a: Therefore, the generating function for the blend, x + a: + 0a: 5 + 5a: _j_ on^.

5 GENERATING FUNCTIONS OF CENTRAL VALUES [Feb. which is precisely the value of. Thus, we see that the limit of the h X generating functions for the (H - L) Ik sequences is precisely the generating function for the maximum values in the rows of the binomial triangle.. GENERATING FUNCTIONS OF THE {H - L)Ik SEQUENCES IN THE TRINOMIAL TRIANGLE Table 4 exhibits the (H -L)/k sequences for k = 4 to k = partitions. The generating function of the maximum values in the trinomial triangle is l//l - x - x. Table 4 (H - L)/k SEQUENCES FOR k = 4 TO k = k = = _ Consider the relation F n + = F n+ + F n9 which is expressed by the equation = 0. and l i m F L n+l jn L F n + F n+ + F n + thus L = + L 9 Fn L = L +, L - L - = 0. Next consider the relation S n + =S n + - %S n + i + % S n > expressed by the equation K - K + xk - x = 0, or in factored form (K - x)(k - ( - x)k + x ) = 0.

6 979] GENERATING FUNCTIONS OF CENTRAL VALUES S n+l lim - = L 9 n + o n which is the root of the maximum modulus by Gauss f s theorem. Further, S n -i lim " S n + - x S n. A - x - a? ' which i s t h e g e n e r a t i n g function of t h e maximum v a l u e s of t h e t r i n o m i a l t r i - a n g l e. Assume where and The r o o t s of Sn S n _ ± S n + - X S n _ ^ ± _ ^ L* - ^ L - lim J_J-l.ll q, S. _, ^n + ^n L Sn-i = lim n -* ^ S-n- X - X + a?z - x = 0 and - x - A - x - x X, ^ 5 - x - A - x - x _ x + A - a? - a? The dominant r o o t i s r, which i s L. Thus, and T h e r e f o r e, r _ ( - x) - x + ( - x)a - x - x ~ r ( - a?) - 4a? + ( - x)v± - x - x L X 7y L - x + A - a; - a? L - x ( - x) - 4a? + ( - x)a - a: - a; - x + A - a? - a? " ( - a? - a? ) + ( - x)a - a? - a? / l - a? - a?

7 4 GENERATING FUNCTIONS OF CENTRAL VALUES [Feb. The generating functions for the odd cases were found to have the form S n (x) S n+ (x) - x S n _ (x) The polynomials i S n with the recurrence S n + = S n+ - xs n+ + x S n are listed as follows: 5 i = 5 = 5 = - x S fy = - x + x S 5 = S s = - kx + x + x - x h S 7 = - 5x + x + x - bx h + x s etc. Thus, the generating functions for N = n 4- are as follows: * * s - x Si s 7 is s h - x z S z Sn is is N = 5 is Ss Ss s 7 - x S Ss - x S, t Se - x S 5 Sv - x - x - x - x - x - x - x + x x + x + x kx + ;c + 5x - x h - x 5 - kx + x + x - x h 5x + 5x + x - lx h - x 5 + x & - 5x + x + x - x h + x s ;r - x + 9x + 5x - 5 ^ + 5x Before the generating functions for the even cases are given, the Lucas, L n (x), and Fibonacci, F n (x), polynomials for the factor K - ( - x)k + x will be derived. The Lucas and Fibonacci polynomials are defined: L n (x) = a n (x) + b n (x) F n (x) = a n (x) - b n (x)/a(x) -?(a?) where a and b are the roots of the polynomial equation K - A(x)K + B(x) = 0. The recurrence relation for the Lucas polynomials is L n+ (x) = ( - x)l n+ (x) - x L n (x).

8 979] GENERATING FUNCTIONS OF CENTRAL VALUES.e polynomials are L 0 = L = - x Li = "" X X L = - x + x L h = - kx + a; + 4x L 5 = - 5x + 5x + 5x etc. - X* - 5x h -- x 5 The recurrence relation for the. Fibonacci polynomials is The polynomials are F n + (x) = ( " x)f n+ {x) F 0 = 0 F x = F = - x F = - x F k = - x + x + x F 5 = - 4x + x - x - x k etc. - x z F n (x) The generating functions for N = kn were found to have the fo F n and the generating functions for N = kn + were found to be F n ~ * Fn-l L n - x L n _i They are listed below. Fi N = 4 is LY I - x i is - F, - x F n L - x L Q - x - x. ^ i - x is L i _ x - x F ~ XF - X N = 0 is - X L - x L l

9 GENERATING FUNCTIONS OF CENTRAL VALUES [Feb. X X ' N- is = - - x + x, 7. F ~ * F \ - x - x + x N =.4 is = L - ;r L - x - x + 4# + x h _, r. r^ - x + x - + ar JV = is = ^ - 4a: + ar + 4x d - x* Lastly, we show F i Fy, - x F w. lim = n / l - x - x and lim L n - x L n. A - x - x We r e c a l l t h a t t h e e q u a t i o n has r o o t s i^ = K - ( - x)k + x = 0 ( - x) + / ( l x, _, ( - x) - / ( l - x ) - 4a: : ~ - and K 0 We d e f i n e n n and F = * " * Tn, 7,n L n = K r ; + K; Note t h a t Thus, K l - K = A - x - x. Now, s i n c e Z > Z, L " (K - K ){K n x + K") + K l l < «! - * > Jf \" - W. ^_ l i m = = L. ikh We use this result to prove the second limit = L. F n X F n~l l> n -l F n-l X L n- L - X L lim L n L n-l L - X X L n _! L n-

10 979] GENERATING FUNCTIONS OF CENTRAL VALUES 7 Fr, F, n- F n -i F n _ L n _ = L 4. GENERATING FUNCTIONS OF THE (H - L)/k SEQUENCES IN A MULTINOMIAL TRIANGLE We challenge the reader to find the generating functions of the (H - L)/k sequences in the quadrinomial triangle. We surmise that the limits would be the generating functions of the central values in Pascal's quadrinomial triangle. REFERENCES. John L. Brown, Jr., & V. E. Hoggatt, Jr., "A Primer for the Fibonacci Numbers, Part XVI: The Central Column Sequence," The Fibonacci Quarterly, No. (978):4.. Michel Y. Rondeau, "The Generating Functions for the Vertical Columns of (7/ + l)-nomial Triangles" (Master's thesis, San Jose State University, San Jose, California, May 978).. Claudia R. Smith, "Sums of Partition Sets in the Rows of Generalized Pascal's Triangles" (Master's thesis, San Jose State - University, San Jose, California, May 978). SOLUTION OF V * ) = \x+i) I N TERMS OF FIBONACCI NUMBERS JAMES C. OWINGS, J R. University of Maryland, College Park, MD 074 In [, pp. ] we solved t h e Diophantine e q u a t i o n P J = ( ^ J and found t h a t (x,y) i s a s o l u t i o n i f f for some n >_ 0, (x +,/ + ) = ( /(4fc + ), J /(4fc + )), \ k - 0 k = 0 / where / ( 0 ) = ^ f{±) = ^ f{n + ) = f(n) + f(n + }. We show h e r e t h a t (x,y) i s a s o l u t i o n i f f for some n >_ 0, (x + l.z/ + ) = (f(n + l)f{n + ), f(n + )f(n + ) ), i n c i d e n t a l l y d e r i v i n g t h e i d e n t i t i e s f(n + l)f(n + ) = ] T / ( 4 k + ), k = o n f(n + )f(n + ) = ] T /(4fe + ). k = 0

COMPOSITIONS AND FIBONACCI NUMBERS

COMPOSITIONS AND FIBONACCI NUMBERS V. E. HOGGATT, JR., and D. A. LIND San Jose State College, San Jose, California and University of Cambridge, England 1, INTRODUCTION A composition of n is an ordered