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 A276099 #19 Mar 07 2025 06:29:10
%S A276099 1,96,24310,7028736,2149374150,678057476096,218191487357116,
%T A276099 71184392021606400,23459604526110889542,7791432263086689484800,
%U A276099 2603575153867220801823060,874329826463740757819785216,294822072977645830504963830300
%N A276099 a(n) = (9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!).
%C A276099 Let a > b be nonnegative integers. The ratio of factorials (2*a*n)!*(b*n)!/( (a*n)!*(2*b*n)!*((a - b)*n)! ) is known to be an integer for all integer n >= 0 (see, for example, Bober, Theorem 1.1). We have the companion result: 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 = 4, b = 2. Other cases include A091496 (a = 2, b = 0), A091527 (a = 1, b = 0), A262732 (a = 2, b = 1), A262733 (a = 3, b = 2) and A276098 (a = 3, b = 1).
%D A276099 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A276099 Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%H A276099 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
%F A276099 a(n) = Sum_{k = 0..2*n} binomial(9*n, k)*binomial(7*n - k - 1, 2*n - k).
%F A276099 a(n) = Sum_{k = 0..n} binomial(14*n, 2*n - 2*k)*binomial(5*n + k - 1, k).
%F A276099 a(n) ~ 1/sqrt(4*Pi*n) * (3^18/5^5)^(n/2).
%F A276099 O.g.f. A(x) = Hypergeom([17/18, 13/18, 11/18, 7/18, 5/18, 1/18, 5/6, 1/6], [9/10, 7/10, 3/10, 1/10, 3/4, 1/4, 1/2], (3^18/5^5)*x^2) + 96*x*Hypergeom([13/9, 11/9, 10/9, 8/9, 7/9, 5/9, 4/3, 2/3], [7/5, 6/5, 4/5, 3/5, 5/4, 3/4, 3/2], (3^18/5^5)*x^2).
%F A276099 a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^9/(1 - x)^5.
%F A276099 It follows that the o.g.f. A(x) 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 A276099 Let F(x) = (1/x)*Series_Reversion( x*sqrt((1 - x)^5/(1 + x)^9) ) and put G(x) = 1 + x*d/dx(log(F(x))). Then A(x^2) = (G(x) + G(-x))/2.
%p A276099 seq(simplify((9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!)), n = 0..20);
%t A276099 Table[((9n)!(5/2 n)!)/((9 n/2)!(5n)!(2n)!),{n,0,15}] (* _Harvey P. Dale_, May 21 2024 *)
%o A276099 (Python)
%o A276099 from math import factorial
%o A276099 from sympy import factorial2
%o A276099 def A276099(n): return int((factorial(9*n)*factorial2(5*n)<<(n<<1))//factorial2(9*n)//factorial(5*n)//factorial(n<<1)) # _Chai Wah Wu_, Aug 10 2023
%Y A276099 Cf. A091496, A091527, A262732, A262733, A276098.
%K A276099 nonn,easy
%O A276099 0,2
%A A276099 _Peter Bala_, Aug 22 2016