A273629 - OEIS

%I #19 Nov 18 2024 22:31:42

%S 1,72,18360,5920200,2118223800,803927196072,316938365223480,

%T 128313095514575400,52976845635264939960,22204947580777261872000,

%U 9418997650746914743158360,4034374193416822645489549632,1741969558937890710303111545400

%N a(n) = (9*n)!/((7*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(4*n + k,n)*binomial(5*n - k,n) = (-1)^m*a(m) for n = 2*m. The sum vanishes for n odd. For similar results see A001451, A006480 and A273628.

%C Note the related sums:

%C Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(4*n - k,n)*binomial(5*n - k,n) = binomial(2*n,n)*binomial(4*n,n) = A000984(n)*A005810(n);

%C Sum_{k = 0..2*n} (-1)^k*binomial(n,k)*binomial(4*n + k,n)*binomial(5*n + k,n) = Sum_{k = 0..2*n} (-1)^k*binomial(n,k)*binomial(4*n - k,n)*binomial(5*n - k,n) = binomial(2*n,n) = A000984(n).

%C Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(4*n + k,n)*binomial(5*n - k,n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(4*n - k,n)*binomial(5*n + k,n) = (-1)^n*binomial(2*n,n) = (-1)^n*A000984(n).

%F a(n) = (9*n)!/((7*n)!*n!^2) = binomial(9*n,2*n)* binomial(2*n,n).

%F a(n) = binomial(8*n,n)*binomial(9*n,n) = A004381(n)*A169958(n).

%F a(n) = [x^n](1 + x)^(8*n) * [x^n] (1 + x)^(9*n).

%F It appears that a(n) = [x^n] F(x)^(72*n), where F(x) = 1 + x + 56*x^2 + 7700*x^3 + 1422008*x^4 + 307144278*x^5 + 73118586828*x^6 + ... has all integer coefficients. Cf. A273628 and A008979.

%F Recurrence: 7*n^2*(7*n - 1)*(7*n - 2)*(7*n - 3)*(7*n - 4)*(7*n - 5)*(7*n - 6)*a(n) = 9*(9*n - 1)*(9*n - 2)*(9*n - 3)*(9*n - 4)*(9*n - 5)*(9*n - 6)*(9*n - 7)*(9*n - 8)*a(n-1).

%F a(n) ~ 3^(18*n+1)*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(8*n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). - _Peter Bala_, Oct 15 2024

%p seq((9*n)!/((7*n)!*n!^2), n = 0..20);

%t Table[Factorial[9 n] / (Factorial[7 n] Factorial[n]^2), {n, 0, 20}] (* _Vincenzo Librandi_, Jul 17 2016 *)

%o (Magma) [Factorial(9*n)/(Factorial(7*n)*Factorial(n)^2): n in [0..40]]; // _Vincenzo Librandi_, Jul 17 2016

%Y Cf. A000984, A001451, A004381, A005810, A006480, A008979, A169958, A245086, A273628.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 15 2016