A364520 a(n) = (7*n)!*(n/2)!/((7*n/2)!*(n)!*(3*n)!).
1, 64, 12012, 2621440, 608435100, 146028888064, 35794148650260, 8901646138474496, 2237242000722428700, 566823049100850626560, 144520856111821003326512, 37036782455383679028953088, 9531607276865293630675462980, 2461693334077582876433071472640
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..400
Programs
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Maple
seq( simplify((7*n)!*(n/2)!/((7*n/2)!*(n)!*(3*n)!)), n = 0..15);
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Mathematica
A364520[n_]:=(7n)!(n/2)!/((7n/2)!n!(3n)!);Array[A364520,15,0] (* Paolo Xausa, Oct 05 2023 *)
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Python
from math import factorial from sympy import factorial2 def A364520(n): return int((factorial(7*n)*factorial2(n)<<(3*n))//(factorial2(7*n)*factorial(n)*factorial(3*n))) # Chai Wah Wu, Aug 13 2023
Formula
a(n) = [x^(3*n)] ( (1 + x)^7/(1 - x) )^n.
a(n) = Sum_{j = 0..3*n} binomial(7*n, j)*binomial(4*n-j-1, 3*n-j).
a(n) = binomial(4*n-1, 3*n) * hypergeom([-7*n, -3*n], [1 - 4*n], -1) for n >= 2.
a(n) ~ c^n * 1/sqrt(6*Pi*n) where c = (14/3)^3*sqrt(7).
P-recursive: a(n) = 448*(7*n-1)*(7*n-3)*(7*n-5)*(7*n-9)*(7*n-11)*(7*n-13)/(3*n*(3*n-1)*(3*n-2)*(3*n-3)*(3*n-4)*(3*n-5)) * a(n-2) with a(0) = 1 and a(1) = 64.
The generating function is an algebraic function over the field of rational functions Q(x).
Comments