login
A368513
a(n) = (6*n+1)!/(n!*(2*n)!*(3*n+1)!).
1
1, 105, 25740, 7759752, 2574148500, 902522205585, 328074738591600, 122332313750680800, 46485667563689950596, 17924037162454524601500, 6991900927489809108938160, 2753354160571011216583946400, 1092796344333659321191117573200, 436609643814534385348768088729640, 175431288302508215774213129529432000
OFFSET
0,2
FORMULA
G.f.: hypergeometric3F2([1/3, 5/6, 7/6], [1, 4/3], 432*z).
E.g.f.: hypergeometric3F3([1/3, 5/6, 7/6], [1, 1, 4/3], 432*z).
a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(3)/(12*Pi*x^(2/3)) - gamma(2/3)*gamma(5/6)*sqrt(3)*hypergeometric3F2([1/2, 5/6, 5/6], [2/3, 3/2], x/432)/(432*Pi^(5/2)*x^(1/6)) + x^(1/6)*hypergeometric3F2([5/6, 7/6, 7/6], [4/3, 11/6], x/432)/(12960*sqrt(Pi)*gamma(2/3)*gamma(5/6)). W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.0007368284. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem.
The shape of W(x) in the above integral representation of a(n) resembles very much the shape of the corresponding W(x) in A113424.
MAPLE
seq((6*n+1)!/(n!*(2*n)!*(3*n+1)!), n=0..14)
CROSSREFS
Cf. A113424.
Sequence in context: A082368 A001546 A111647 * A295463 A339847 A145621
KEYWORD
nonn
AUTHOR
Karol A. Penson, Dec 28 2023
STATUS
approved