A380212 Expansion of e.g.f. exp( 1/(1-2*x)^(3/2) - 1 ).
1, 3, 24, 267, 3771, 64188, 1273599, 28784997, 728619516, 20389690953, 624380711769, 20749726230192, 743217114278241, 28526465892902643, 1167521852585583504, 50735768950040355747, 2332267950561718237011, 113040281313704744222148, 5759890462485002871029439
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[3^k * 2^(n-k) * Abs[StirlingS1[n,k]] * BellB[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-2*x)^(3/2)-1)))
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * |Stirling1(n,k)| * Bell(k).
a(n) = (1/e) * (-2)^n * n! * Sum_{k>=0} binomial(-3*k/2,n)/k!.
a(n) ~ 3^(1/5) * 5^(-1/2) * 2^(n + 3/10) * n^(n - 1/5) * exp(-1 + 2^(1/5)*3^(4/5)*n^(1/5)/4 + 5*2^(3/5)*3^(2/5)*n^(3/5)/6 - n) * (1 + 2^(4/5)*3^(1/5)/(10*n^(1/5))). - Vaclav Kotesovec, Jan 23 2025