A344130 Expansion of Product_{k>=1} (1 + x^k)^binomial(2*k,k).

1, 2, 7, 32, 131, 562, 2383, 10124, 42916, 181844, 769246, 3250388, 13716377, 57812466, 243382957, 1023463628, 4299199426, 18040918456, 75632083258, 316774424568, 1325591467994, 5542462776932, 23155074355078, 96661979245880, 403223735948096, 1680858909265768

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (-1)^k * (1 - 1/sqrt(1 - 4^(1-k)))/k = -0.0680700790487788003241709878640131435678833005035785286095068...

Mathematica

nmax = 25; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[Sum[-(-1)^j * (1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x]

Crossrefs

Cf. A000984, A304787, A344108.

Sequence in context: A271817 A015655 A320682 * A327214 A334855 A334854

Adjacent sequences: A344127 A344128 A344129 * A344131 A344132 A344133

Keyword

nonn

Author

Vaclav Kotesovec, May 10 2021

Status

approved