A380215 Expansion of e.g.f. exp( (1+3*x)^(2/3) - 1 ).
1, 2, 2, 4, -12, 152, -2056, 34064, -663792, 14890656, -378083936, 10721383488, -335898007232, 11523599785856, -429685396446848, 17303743585216768, -748494039183318784, 34612915914568045056, -1704065501541830102528, 88989595986614229074944, -4913365756826406035999744
Offset: 0
Keywords
Programs
-
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1+3*x)^(2/3)-1)))
Formula
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling1(n,k) * Bell(k).
a(n) = (1/e) * 3^n * n! * Sum_{k>=0} binomial(2*k/3,n)/k!.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380261.
a(n) ~ (-1)^(n+1) * 2^(3/2) * sqrt(Pi) * 3^(n-1) * n^(n - 7/6) / (Gamma(1/3) * exp(n+1)). - Vaclav Kotesovec, Jan 19 2025