A134772 Let f(m, n) = 2^(-m) * Sum_{j=0..n} (-1)^j*m!*n!*(2*m-2*j)!/(j!*(m-j)! *(n-j)!*6^(n-j)) then a(n) = f(2n,n).
1, 0, 28, 14400, 27165600, 134289792000, 1445549490000000, 29865588836219136000, 1081114481157129619200000, 64007711015400701105356800000, 5873237165016878140678626432000000, 799901135455942846519287494400000000000, 156064894765355001368149078831725782016000000
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..128
Programs
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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
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Mathematica
Table[((4*n)!/(24)^n)*Hypergeometric1F1[-n, 1/2-2*n, -3/2], {n,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
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