A380259 - OEIS

%I #14 Jan 23 2025 05:34:01

%S 1,1,6,51,561,7566,120711,2221311,46269126,1075249881,27560477331,

%T 771948530046,23446574573841,767288588019201,26905482997736526,

%U 1006166248423254171,39962774633459923881,1679677496419394133846,74471142324541556576151

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F 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.

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

%F 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

%t 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 *)

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

%Y Cf. A049377, A380212.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 18 2025