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URL: https://oeis.org/A273629

⇱ A273629 - OEIS

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A273629
a(n) = (9*n)!/((7*n)!*n!^2).
3
1, 72, 18360, 5920200, 2118223800, 803927196072, 316938365223480, 128313095514575400, 52976845635264939960, 22204947580777261872000, 9418997650746914743158360, 4034374193416822645489549632, 1741969558937890710303111545400
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(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.
Note the related sums:
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);
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).
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).
FORMULA
a(n) = (9*n)!/((7*n)!*n!^2) = binomial(9*n,2*n)* binomial(2*n,n).
a(n) = binomial(8*n,n)*binomial(9*n,n) = A004381(n)*A169958(n).
a(n) = [x^n](1 + x)^(8*n) * [x^n] (1 + x)^(9*n).
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.
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).
a(n) ~ 3^(18*n+1)*7^(-7*n-1/2)/(2*Pi*n). - Ilya Gutkovskiy, Jul 15 2016
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
MAPLE
seq((9*n)!/((7*n)!*n!^2), n = 0..20);
MATHEMATICA
Table[Factorial[9 n] / (Factorial[7 n] Factorial[n]^2), {n, 0, 20}] (* Vincenzo Librandi, Jul 17 2016 *)
PROG
(Magma) [Factorial(9*n)/(Factorial(7*n)*Factorial(n)^2): n in [0..40]]; // Vincenzo Librandi, Jul 17 2016
(Python)
from math import comb
def A273629(n): return comb(9*n, m:=n<<1)*comb(m, n) # Chai Wah Wu, Mar 26 2026
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 15 2016
STATUS
approved