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A171605 Coefficients of Hankel moment polynomials for c=1/2:f(a,b) = Gamma[a + b]/Gamma[a] p(x,n)=Sum[Binomial(n, k)*(f(c, n)/(f(c, n - k)*f(c, k)))*x^k, {k, 0, n}].

%I A171605 #4 Jul 22 2025 07:31:47
%S A171605 1,1,-1,-1,1,1,2,-17,28,-17,2,1,1,9,-60,116,-66,-66,116,-60,9,1,1,20,
%T A171605 -126,196,239,-1240,1820,-1240,239,196,-126,20,1,1,35,-195,15,2205,
%U A171605 -7001,9785,-4845,-4845,9785,-7001,2205,15,-195,35,1,1,54,-231,-880
%N A171605 Coefficients of Hankel moment polynomials for c=1/2:f(a,b) = Gamma[a + b]/Gamma[a] p(x,n)=Sum[Binomial(n, k)*(f(c, n)/(f(c, n - k)*f(c, k)))*x^k, {k, 0, n}].
%C A171605 Row sums are zero except for n=0.
%C A171605 Other Hankel moments are:
%C A171605 c = 2 : A001263, Narayana
%C A171605 ;c = 1 : A008459. binomial squared.
%D A171605 Philip Feinsilver and Rene Schott, Algebraic Structure and Operator Calculus; Volume I: Representations and Probability Theory,Kluwer,London,1993, ISBN 0-7923-2116-2,page 7
%F A171605 f(a,b) = Gamma[a + b]/Gamma[a]
%F A171605 p(x,n)=Sum[Binomial(n, k)*(f(c, n)/(f(c, n - k)*f(c, k)))*x^k, {k, 0, n}]
%e A171605 {1},
%e A171605 {1, -1, -1, 1},
%e A171605 {1, 2, -17, 28, -17, 2, 1},
%e A171605 {1, 9, -60, 116, -66, -66, 116, -60, 9, 1},
%e A171605 {1, 20, -126, 196, 239, -1240, 1820, -1240, 239, 196, -126, 20, 1},
%e A171605 {1, 35, -195, 15, 2205, -7001, 9785, -4845, -4845, 9785, -7001, 2205, 15, -195, 35, 1},
%e A171605 {1, 54, -231, -880, 8052, -21912, 22276, 20976, -95634, 134596, -95634, 20976, 22276, -21912, 8052, -880, -231, 54, 1},
%e A171605 {1, 77, -182, -3094, 19929, -43043, -10920, 268568, -665406, 810810, -376740, -376740, 810810, -665406, 268568, -10920, -43043, 19929, -3094, -182, 77, 1},
%e A171605 {1, 104, 20, -7272, 37762, -42120, -270140, 1299080, -2608913, 2193808, 1776424, -7637904, 10518300, -7637904, 1776424, 2193808, -2608913, 1299080, -270140, -42120, 37762, -7272, 20, 104, 1},
%e A171605 {1, 135, 459, -13923, 55998, 59058, -1151070, 4053582, -6097509, -1814851, 27460881, -59839065, 67546644, -30260340, -30260340, 67546644, -59839065, 27460881, -1814851, -6097509, 4053582, -1151070, 59058, 55998, -13923, 459, 135, 1},
%e A171605 {1, 170, 1235, -23180, 59565, 411502, -3254225, 8979400, -4878915, -43714630, 159116983, -271019060, 204302345, 149970990, -623782445, 847660528, -623782445, 149970990, 204302345, -271019060, 159116983, -43714630, -4878915, 8979400, -3254225, 411502, 59565, -23180, 1235, 170, 1}
%t A171605 f[a_, b_] = Gamma[a + b]/Gamma[a]
%t A171605 c = 1/2;
%t A171605 p[x_, n_] = Sum[Binomial[n, k]*(f[c, n]/(f[ c, n - k]*f[c, k]))*x^k, {k, 0, n}]
%t A171605 Table[CoefficientList[p[x, n], x], {n, 0, 10}]
%t A171605 Flatten[%]
%Y A171605 A001263, A008459
%K A171605 sign,uned
%O A171605 0,7
%A A171605 _Roger L. Bagula_, Dec 12 2009