A382449 Expansion of e.g.f. exp( x/(1-2*x)^(3/2) ).

1, 1, 7, 64, 745, 10576, 177121, 3414622, 74389729, 1805424040, 48264466321, 1408241206186, 44508262018177, 1514115583435924, 55142123112150985, 2139885098048098486, 88128888655032851521, 3838126991973342097072, 176206944426651875454049

Offset: 0

Author

Seiichi Manyama, Apr 03 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A001879, A362204.

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^(x/(1 - 2*x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 13 2025 *)

PARI

a(n) = n!*sum(k=0, n, 2^(n-k)*binomial(n+k/2-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * binomial(n+k/2-1,n-k)/k!.

a(n) ~ 3^(1/5) * 2^(n + 1/10) * exp(-3^(-1/5)*2^(-13/5)*n^(1/5) + 5*3^(-3/5)*2^(-4/5)*n^(3/5) - n) * n^(n - 1/5) / sqrt(5). - Vaclav Kotesovec, Apr 13 2025