OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..128
FORMULA
a(n) = 4^(-n) * Sum_{j=0..n} (-1)^j*(2*n)!*n!*(4*n-2*j)!/(j!*(2*n-j)! *(n-j)!*6^(n-j)).
From G. C. Greubel, Oct 12 2023: (Start)
a(n) = ((4*n)!/(24)^n) * Sum_{j=0..n} b(n,j)*b(2*n,j)(-6)^j/(b(2*j,j) * b(4*n,2*j)), where b(x,y) = binomial(x,y).
a(n) = ((4*n)!/(24)^n) * Hypergeometric1F1([-n], [1/2 -2*n], -3/2).
Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = 1/sqrt(1+x) * Hypergeometric2F0([1/4, 3/4]; --; 8*x/(3*(1+x)^2)). (End)
a(n) ~ sqrt(Pi) * 2^(5*n + 3/2) * n^(4*n + 1/2) / (3^n * exp(4*n + 3/4)). - Vaclav Kotesovec, Oct 21 2023
MATHEMATICA
Table[((4*n)!/(24)^n)*Hypergeometric1F1[-n, 1/2-2*n, -3/2], {n, 0, 30}] (* G. C. Greubel, Oct 12 2023 *)
PROG
(Magma)
B:=Binomial;
A134772:= func< n | Factorial(4*n)/(24)^n *(&+[B(n, j)*B(2*n, j)*(-6)^j/(Factorial(j)*B(2*j, j)*B(4*n, 2*j)) : j in [0..n]]) >;
[A134772(n): n in [0..30]]; // G. C. Greubel, Oct 12 2023
(SageMath)
def A134772(n): return (factorial(4*n)/(24)^n)* simplify(hypergeometric([-n], [1/2-2*n], -3/2))
[A134772(n) for n in range(31)] # G. C. Greubel, Oct 12 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 18 2009
STATUS
approved