A262733 a(n) = (1/n!) * (7*n)!/(7*n/2)! * (5*n/2)!/(5*n)!.

1, 12, 286, 7680, 217350, 6336512, 188296108, 5670567936, 172459427910, 5284842700800, 162922160580036, 5047099485847552, 156983503897469340, 4899363753956474880, 153349672416272587800, 4811846645261721927680, 151316978279502571401798, 4767566079229070105640960

Offset

0,2

Comments

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.

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

References

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Links

Table of n, a(n) for n=0..17.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Peter Bala, Some integer ratios of factorials

Formula

a(n) = [x^n] ( (1 + x)^7/(1 - x)^5 )^n.

a(n) = Sum_{i = 0..n} binomial(7*n,i)*binomial(6*n-i-1,n-i).

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).

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.

a(n) ~ 2^n*5^(-5*n/2)*7^(7*n/2)/sqrt(2*Pi*n). - Ilya Gutkovskiy, Jul 31 2016

From Peter Bala, Aug 22 2016: (Start)

a(n) = Sum_{k = 0..floor(n/2)} binomial(12*n,n - 2*k) * binomial(5*n + k - 1,k).

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).

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)

Maple

a := n -> 1/n! * (7*n)!/GAMMA(1 + 7*n/2) * GAMMA(1 + 5*n/2)/(5*n)!:
seq(a(n), n = 0..18);

Mathematica

Table[1/n!*(7 n)!/(7 n/2)!*(5 n/2)!/(5 n)!, {n, 0, 17}] (* Michael De Vlieger, Oct 04 2015 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(7*n, k)*binomial(6*n-k-1, n-k));
vector(30, n, a(n-1)) \\ Altug Alkan, Oct 03 2015
(Python)
from math import factorial
from sympy import factorial2
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

Crossrefs

Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A211421 (k = 6), A262739, A276098, A276099.

Sequence in context: A154669 A079519 A275007 * A296736 A077424 A275562

Adjacent sequences: A262730 A262731 A262732 * A262734 A262735 A262736

Keyword

nonn,easy

Author

Peter Bala, Sep 29 2015

Status

approved