A380259 Expansion of e.g.f. exp( (1/(1-2*x)^(3/2) - 1)/3 ).

1, 1, 6, 51, 561, 7566, 120711, 2221311, 46269126, 1075249881, 27560477331, 771948530046, 23446574573841, 767288588019201, 26905482997736526, 1006166248423254171, 39962774633459923881, 1679677496419394133846, 74471142324541556576151

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, Bell Polynomial.

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!).

a(n) ~ 2^(n + 3/10) * n^(n - 1/5) * exp(-1/3 + 2^(1/5)*n^(1/5)/4 + 5*2^(3/5)*n^(3/5)/6 - n) / sqrt(5) * (1 + 2^(4/5) / (30 * n^(1/5))). - Vaclav Kotesovec, Jan 23 2025

Mathematica

Table[Sum[3^k * 2^(n-k) * Abs[StirlingS1[n, k]] * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *)

Prog

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

Crossrefs

Cf. A049377, A380212.

Sequence in context: A124565 A057817 A000405 * A113352 A063169 A346667

Adjacent sequences: A380256 A380257 A380258 * A380260 A380261 A380262

Keyword

nonn,new

Author

Seiichi Manyama, Jan 18 2025

Status

approved