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

%I A364304 #27 Oct 07 2023 06:59:07
%S A364304 1,54,10296,2484000,665091000,188907932304,55737530929080,
%T A364304 16888537352985408,5218680924762089400,1637124203403474142500,
%U A364304 519752205290081232622296,166620892958456148158454144,53846423260084127389865311800
%N A364304 a(n) = (7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2).
%C A364304 Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! = gamma(1 + 9*n/2).
%H A364304 Paolo Xausa, <a href="/A364304/b364304.txt">Table of n, a(n) for n = 0..300</a>
%F A364304 a(n) = Sum_{k = 0..n} binomial(7*n, n - k)^2 * binomial(5*n + k - 1, k).
%F A364304 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).
%F A364304 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)
%F A364304 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.
%F A364304 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.
%F A364304 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.
%F A364304 a(p) == a(1) (mod p^3).
%F A364304 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.
%F A364304 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.
%F A364304 a(n) ~ c^n * 3*sqrt(7)/(14*Pi*n), where c = (3^9)/(5^3) * sqrt(5).
%p A364304 seq( simplify((7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2)), n = 0..12);
%t A364304 A364304[n_]:=(7n)!(9n/2)!(5n/2)!/((5n)!(7n/2)!^2n!^2);Array[A364304,15,0] (* _Paolo Xausa_, Oct 06 2023 *)
%o A364304 (Python)
%o A364304 from math import factorial
%o A364304 from sympy import factorial2
%o A364304 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
%Y A364304 Row 7 of A364303.
%Y A364304 Cf. A275652, A275654.
%K A364304 nonn,easy
%O A364304 0,2
%A A364304 _Peter Bala_, Jul 21 2023