This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091496 #35 May 05 2025 02:54:10
%S A091496 1,16,630,28672,1385670,69206016,3528923580,182536110080,
%T A091496 9540949030470,502682972323840,26651569523959380,1420217179365703680,
%U A091496 75998432812419471900,4081125953526124511232,219813190240007470094520,11869871068877664049692672,642409325786050322446410310
%N A091496 a(n) = ((5*n)!/(n!*(2*n)!))*(Gamma(1+n/2)/Gamma(1+5*n/2)).
%C A091496 Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for all integer n >= 0. This is the case a = 2, b = 0. - _Peter Bala_, Aug 28 2016
%D A091496 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A091496 G. C. Greubel, <a href="/A091496/b091496.txt">Table of n, a(n) for n = 0..500</a>
%H A091496 Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%F A091496 n*(n-1)*(2*n-1)*(2*n-3)*a(n) = 20*(5*n-1)*(5*n-3)*(5*n-7)*(5*n-9)*a(n-2).
%F A091496 From _Peter Bala_, Aug 22 2016: (Start)
%F A091496 a(n) = Sum_{k = 0..2*n} binomial(5*n, k)*binomial(3*n - k - 1, 2*n - k).
%F A091496 a(n) = Sum_{k = 0..n} binomial(6*n, 2*n - 2*k)*binomial(n + k - 1, k).
%F A091496 a(n) ~ 5^(5*n/2)/(2*sqrt(Pi*n)).
%F A091496 O.g.f. A(x) = Hypergeom([9/10, 7/10, 3/10, 1/10], [3/4, 1/2, 1/4], 3125*x^2) + 16*x*Hypergeom([7/5, 6/5, 4/5, 3/5], [5/4, 3/2, 3/4], 3125*x^2).
%F A091496 a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^5/(1 - x). Cf. A061162 and A262732.
%F A091496 It follows that the o.g.f. for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
%F A091496 exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 16*x + 443*x^2 + 15280*x^3 + 591998*x^4 + 24635360*x^5 + 1075884051*x^6 + ... has integer coefficients.
%F A091496 Let F(x) = 1/x*Series_Reversion( x*sqrt((1 - x)/(1 + x)^5) ) and put G(x) = 1 + x*d/dx(log(F(x))). Then A(x) satisfies A(x^2) = (G(x) + G(-x))/2. (End)
%F A091496 O.g.f. denoted by h(x), satisfies algebraic equation of order 10: -800000*x^4 + 81*x^2 - 25000*x^3*h(x) - 25*x^2*(1400000*x^2 - 123)*h(x)^2 + 8*x*(178125*x^2 - 32)*h(x)^3 + (-31250000*x^4 + 22500*x^2 + 4)*h(x)^4 + 32*x*(137500*x^2 - 19)*(3125*x^2 - 1)*h(x)^5 + 12*(3125*x^2 - 1)*(3125*x^2 + 3)*h(x)^6 + 800*x*(3125*x^2 - 1)^2*h(x)^7 + 96*(3125*x^2 - 1)^2*h(x)^8 + 64*(3125*x^2 - 1)^3*h(x)^10 = 0. - _Karol A. Penson_, Apr 30 2025
%t A091496 Table[((5 n)!/(n! (2 n)!)) (Gamma[1 + n/2]/Gamma[1 + 5 n/2]), {n, 0, 14}] (* or *)
%t A091496 Table[Sum[Binomial[6 n, 2 n - 2 k] Binomial[n + k - 1, k], {k, 0, n}], {n, 0, 14}] (* or *)
%t A091496 Table[Sum[Binomial[5 n, k] Binomial[3 n - k - 1, 2 n - k], {k, 0, 2 n}], {n, 0, 14}] (* _Michael De Vlieger_, Aug 28 2016 *)
%o A091496 (PARI) a(n)=16^n*sum(i=0,2*n,binomial(i-1+(n-1)/2,i))
%o A091496 (Python)
%o A091496 from math import factorial
%o A091496 from sympy import factorial2
%o A091496 def A091496(n): return int((factorial(5*n)*factorial2(n)<<(n<<1))//(factorial(n)*factorial(n<<1)*factorial2(5*n))) # _Chai Wah Wu_, Aug 10 2023
%Y A091496 Cf. A061163(n)=a(2n), A061162, A091527, A262732, A262733, A276098, A276099.
%K A091496 nonn
%O A091496 0,2
%A A091496 _Michael Somos_, Jan 15 2004