A273628 a(n) = (7*n)!/((5*n)!*n!^2).

1, 42, 6006, 1085280, 217567350, 46262007792, 10217700004512, 2317454130543552, 536022010184210550, 125863265857621191900, 29909151834298018538256, 7176685161839833601969280, 1735941935586019529116213920, 422752608090008019258722317800

Offset

0,2

Comments

This sequence occurs as the right-hand side of the binomial sum identity Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(3*n + k,n)*binomial(4*n - k,n) = (-1)^m*a(m) for n = 2*m. For similar results see A001451, A006480 and A273629. Note the related sums:

Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(3*n + k,n)*binomial(4*n + k,n) = (-1)^n*(2*n)!*(4*n)!/(n!^3*(3*n)!) = (-1)^n*binomial(2*n,n)*binomial(4*n,n) = (-1)^n*A000984(n)*A005810(n);

Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(3*n - k,n)*binomial(4*n - k,n) = (3*n)!/n!^3 = A006480(n);

Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(3*n + k,n)*binomial(4*n + k,n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(3*n - k,n)*binomial(4*n - k,n) = binomial(2*n,n) = A000984(n);

Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(3*n + k,n)*binomial(4*n - k,n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(3*n - k,n)*binomial(4*n + k,n) = (-1)^n*binomial(2*n,n) = (-1)^n*A000984(n).

Links

Table of n, a(n) for n=0..13.

Formula

a(n) = (7*n)!/((5*n)!*n!^2) = binomial(7*n,2*n)*binomial(2*n,n).

a(n) = binomial(7*n,n)*binomial(6*n,n) = [x^n](1 + x)^(7*n) * [x^n](1 + x)^(6*n).

It appears that a(n) = [x^n] F(x)^(42*n), where F(x) = 1 + x + 30*x^2 + 2280*x^3 + 232715*x^4 + 27800465*x^5 + 3661895341*x^6 + ... has all integer coefficients. Cf. A273629 and A008979.

Recurrence: 5*n^2*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)*a(n) = 7*(7*n - 1)*(7*n - 2)*(7*n - 3)*(7*n - 4)*(7*n - 5)*(7*n - 6)*a(n - 1).

a(n) ~ 5^(-5*n-1/2)*7^(7*n+1/2)/(2*Pi*n). - Ilya Gutkovskiy, Jul 15 2016

Maple

seq((7*n)!/((5*n)!*n!^2), n = 0..20);

Mathematica

Table[(7 n)!/((5 n)! n!^2), {n, 0, 13}] (* or *)
Table[Binomial[7 n, n] Binomial[6 n, n], {n, 0, 13}] (* Michael De Vlieger, Jul 15 2016 *)

Prog

(Magma) [Factorial(7*n) div (Factorial(5*n)*Factorial(n)^2): n in [0..15]]; // Vincenzo Librandi, Jul 16 2016

Crossrefs

Cf. A000984, A001451, A005810, A006480, A008979, A245086, A273629.

Sequence in context: A102967 A091545 A101630 * A005791 A167668 A277665

Adjacent sequences: A273625 A273626 A273627 * A273629 A273630 A273631

Keyword

nonn,easy

Author

Peter Bala, Jul 15 2016

Status

approved