A364304 a(n) = (7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2).

1, 54, 10296, 2484000, 665091000, 188907932304, 55737530929080, 16888537352985408, 5218680924762089400, 1637124203403474142500, 519752205290081232622296, 166620892958456148158454144, 53846423260084127389865311800

Offset

0,2

Comments

Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! = gamma(1 + 9*n/2).

Links

Paolo Xausa, Table of n, a(n) for n = 0..300

Formula

a(n) = Sum_{k = 0..n} binomial(7*n, n - k)^2 * binomial(5*n + k - 1, k).

a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(7*n + k, 7*n - k)*binomial(2*k, k)*binomial(2*n - k, n).

Conjecture: a(n) = Sum_{k = 0..7*n} (-1)^k * binomial(7*n + k, 7*n - k)* binomial(2*k, k)*binomial(2*n - k, n)

a(n) = [x^n] (1 - x)^(2*n) * P(7*n, (1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial.

a(n) = [x^n] G(x)^(9*n), where G(x) = 1 + 6*x + 266*x^2 + 27104*x^3 + 3726380*x^4 + 600232416*x^5 + 106662768380*x^6 + ... appears to have integer coefficients.

exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^9, where F(x) = 1 + 6*x + 590*x^2 + 95468*x^3 + 19200692*x^4 + 4364084760*x^5 + 1072849548644*x^6 + ... appears to have integer coefficients.

a(p) == a(1) (mod p^3).

Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

P-recursive: a(n) = 9*(9*n-2)*(9*n-4)*(9*n-6)*(9*n-8)*(9*n-10)*(9*n-12)*(9*n-14)*(9*n-16)*(7*n-1)*(7*n-3)*(7*n-5)*(7*n-9)*(7*n-11)*(7*n-13)/((7*n-2)*(7*n-4)*(7*n-6)*(7*n-8)*(7*n-10)*(7*n-12)*(5*n-1)*(5*n-3)*(5*n-5)*(5*n-7)*(5*n-9)*n^2*(n-1)) * a(n-2) with a(0) = 1 and a(1) = 54.

a(n) ~ c^n * 3*sqrt(7)/(14*Pi*n), where c = (3^9)/(5^3) * sqrt(5).

Maple

seq( simplify((7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2)), n = 0..12);

Mathematica

A364304[n_]:=(7n)!(9n/2)!(5n/2)!/((5n)!(7n/2)!^2n!^2); Array[A364304, 15, 0] (* Paolo Xausa, Oct 06 2023 *)

Prog

(Python)
from math import factorial
from sympy import factorial2
def A364304(n): return int(factorial(7*n)*factorial2(9*n)*factorial2(5*n)//factorial(5*n)//factorial2(7*n)**2//factorial(n)**2) # Chai Wah Wu, Aug 10 2023

Crossrefs

Row 7 of A364303.

Cf. A275652, A275654.

Sequence in context: A369701 A299949 A228607 * A030254 A085482 A084226

Adjacent sequences: A364301 A364302 A364303 * A364305 A364306 A364307

Keyword

nonn,easy

Author

Peter Bala, Jul 21 2023

Status

approved