A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*x)-1))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
(PARI) a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

Crossrefs

Cf. A052506, A354311.

Cf. A351734, A351737, A354310, A354314.

Sequence in context: A198892 A357790 A205571 * A052826 A358264 A246759

Adjacent sequences: A354309 A354310 A354311 * A354313 A354314 A354315

Keyword

sign

Author

Seiichi Manyama, May 23 2022

Status

approved