A380260 Expansion of e.g.f. exp( ((1+2*x)^(3/2) - 1)/3 ).

1, 1, 2, 3, 9, 6, 111, -573, 7638, -95751, 1450431, -24643134, 468589617, -9843336567, 226448287794, -5662061186949, 152892006728841, -4434211761771978, 137468475061977663, -4536657554920874181, 158788359466681092966, -5875324355407515077439, 229142457698060305226367

Offset: 0

Author

Seiichi Manyama, Jan 18 2025

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Cf. A049425, A380259.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[((1+2x)^(3/2)-1)/3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 29 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(((1+2*x)^(3/2)-1)/3)))

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * Stirling1(n,k) * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * Stirling1(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/3)) * 2^n * n! * Sum_{k>=0} binomial(3*k/2,n)/(3^k * k!).