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 A262733 #43 Aug 09 2025 10:05:07
%S A262733 1,12,286,7680,217350,6336512,188296108,5670567936,172459427910,
%T A262733 5284842700800,162922160580036,5047099485847552,156983503897469340,
%U A262733 4899363753956474880,153349672416272587800,4811846645261721927680,151316978279502571401798,4767566079229070105640960
%N A262733 a(n) = (1/n!) * (7*n)!/(7*n/2)! * (5*n/2)!/(5*n)!.
%C A262733 Sequence terms are given by the coefficient of x^n in the expansion of ( (1 + x)^(k+2)/(1 - x)^k )^n when k = 5. See the cross references for related sequences obtained from other values of k.
%C A262733 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 n >= 0. This is the case a = 3, b = 2. - _Peter Bala_, Aug 28 2016
%D A262733 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A262733 Peter Bala, <a href="/A100100/a100100_1.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%H A262733 Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%F A262733 a(n) = [x^n] ( (1 + x)^7/(1 - x)^5 )^n.
%F A262733 a(n) = Sum_{i = 0..n} binomial(7*n,i) * binomial(6*n-i-1,n-i).
%F A262733 a(n) = 28*(7*n - 1)*(7*n - 3)*(7*n - 9)*(7*n - 11)*(7*n - 13) / ( n*(5*n - 1)*(5*n - 3)*(5*n - 5)*(5*n - 7)*(5*n - 9) ) * a(n-2).
%F A262733 The o.g.f. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 12*x + 215*x^2 + 4564*x^3 + 106442*x^4 + ... has integer coefficients and equals 1/x * series reversion of x*(1 - x)^5/(1 + x)^7. See A262739.
%F A262733 a(n) ~ 2^n*5^(-5*n/2)*7^(7*n/2)/sqrt(2*Pi*n). - _Ilya Gutkovskiy_, Jul 31 2016
%F A262733 From _Peter Bala_, Aug 22 2016: (Start)
%F A262733 a(n) = Sum_{k = 0..floor(n/2)} binomial(12*n,n - 2*k) * binomial(5*n + k - 1,k).
%F A262733 O.g.f.: A(x) = Hypergeom([13/14, 11/14, 9/14, 5/14, 3/14, 1/14], [9/10, 7/10, 3/10, 1/2, 1/10], (2^2*7^7/5^5)*x^2) + 12*x*Hypergeom([10/7, 9/7, 8/7, 6/7, 5/7, 4/7], [7/5, 6/5, 4/5, 3/2, 3/5], (2^2*7^7/5^5)*x^2).
%F A262733 The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^7/(1 - x)^5) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197. (End)
%F A262733 From _Seiichi Manyama_, Aug 09 2025: (Start)
%F A262733 a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(5*n)).
%F A262733 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(7*n,k) * binomial(2*n-k,n-k).
%F A262733 a(n) = Sum_{k=0..n} 2^k * binomial(5*n+k-1,k) * binomial(2*n-k,n-k).
%F A262733 a(n) = 4^n * binomial((7*n-1)/2,n).
%F A262733 a(n) = [x^n] 1/(1-4*x)^((5*n+1)/2).
%F A262733 a(n) = [x^n] (1+4*x)^((7*n-1)/2). (End)
%p A262733 a := n -> 1/n! * (7*n)!/GAMMA(1 + 7*n/2) * GAMMA(1 + 5*n/2)/(5*n)!:
%p A262733 seq(a(n), n = 0..18);
%t A262733 Table[1/n!*(7 n)!/(7 n/2)!*(5 n/2)!/(5 n)!, {n, 0, 17}] (* _Michael De Vlieger_, Oct 04 2015 *)
%o A262733 (PARI) a(n) = sum(k=0, n, binomial(7*n,k)*binomial(6*n-k-1,n-k));
%o A262733 vector(30, n, a(n-1)) \\ _Altug Alkan_, Oct 03 2015
%o A262733 (Python)
%o A262733 from math import factorial
%o A262733 from sympy import factorial2
%o A262733 def A262733(n): return int((factorial(7*n)*factorial2(5*n)<<n)//(factorial2(7*n)*factorial(5*n)*factorial(n))) # _Chai Wah Wu_, Aug 10 2023
%Y A262733 Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A211421 (k = 6), A262739, A276098, A276099.
%K A262733 nonn,easy
%O A262733 0,2
%A A262733 _Peter Bala_, Sep 29 2015