A354324 Expansion of e.g.f. exp( x/8 * (exp(4 * x) - 1) ).

1, 0, 1, 6, 35, 220, 1623, 14294, 144393, 1605384, 19295585, 249938062, 3485830299, 52134346004, 830954821431, 14031857352270, 249956799370193, 4682845238636560, 92038069890608769, 1893193762636115990, 40659808272769543635, 909744112577077608012

Offset

0,4

Links

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

Formula

a(0) = 1; a(n) = (1/2) * Sum_{k=2..n} k * 4^(k-2) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 4^(n-2*k) * Stirling2(n-k,k)/(2^k * (n-k)!).

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/8*(exp(4*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*4^(j-2)*binomial(i-1, j-1)*v[i-j+1])/2); v;
(PARI) a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*stirling(n-k, k, 2)/(2^k*(n-k)!));

Crossrefs

Cf. A354326.

Sequence in context: A354136 A370323 A317409 * A213452 A357834 A000399

Adjacent sequences: A354321 A354322 A354323 * A354325 A354326 A354327

Keyword

nonn

Author

Seiichi Manyama, May 24 2022

Status

approved