A361626 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^2.

1, 3, 17, 139, 1437, 17711, 252133, 4059567, 72779129, 1435276027, 30836352441, 716101686323, 17858449006357, 475653606922599, 13467411746316557, 403708230041927191, 12767545998797849073, 424670548932688771187, 14814998283177691422049

Offset: 0

Author

Seiichi Manyama, Mar 18 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..420

Crossrefs

Cf. A091695, A351767, A361599.

Cf. A000262, A001339, A343884.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^2))

PARI

a(n)=n! * sum(k=0, n, binomial(n+2*k+1,n-k)/k!) \\ Winston de Greef, Mar 18 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+2*k+1,n-k)/k! = Sum_{k=0..n} (n+2*k+1)!/(3*k+1)! * binomial(n,k).

a(n) ~ 3^(5/8) * exp(-1/27 - 3^(3/4)*n^(1/4)/72 + sqrt(3*n)/6 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 3/8) / 6 * (1 + 63037 * 3^(1/4)/(69120 * n^(1/4))). - Vaclav Kotesovec, Mar 29 2023