A273628 - OEIS

%I #17 Oct 16 2024 09:22:56

%S 1,42,6006,1085280,217567350,46262007792,10217700004512,

%T 2317454130543552,536022010184210550,125863265857621191900,

%U 29909151834298018538256,7176685161839833601969280,1735941935586019529116213920,422752608090008019258722317800

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

%C 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:

%C 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);

%C 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);

%C 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);

%C 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).

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

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

%F 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.

%F 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).

%F a(n) ~ 5^(-5*n-1/2)*7^(7*n+1/2)/(2*Pi*n). - _Ilya Gutkovskiy_, Jul 15 2016

%F a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k) * A108625(6*n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). - _Peter Bala_, Oct 15 2024

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

%t Table[(7 n)!/((5 n)! n!^2), {n, 0, 13}] (* or *)

%t Table[Binomial[7 n, n] Binomial[6 n, n], {n, 0, 13}] (* _Michael De Vlieger_, Jul 15 2016 *)

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

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

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 15 2016