A368317 Expansion of e.g.f. exp(4*x) / (2 - exp(x)).

1, 5, 27, 161, 1083, 8345, 74067, 754241, 8726283, 113375465, 1635899907, 25961939921, 449464541883, 8429731963385, 170261482711347, 3684531041231201, 85050474868523883, 2085932272336772105, 54168554611721580387, 1484825397108091268081

Offset: 0

Author

Seiichi Manyama, Dec 21 2023

Keywords

• nonn

Crossrefs

Cf. A000670, A000629, A007047, A259533.

Programs

PARI

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=1) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

Formula

a(n) = 4^n + Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = 16*A000670(n) - (3^n + 2*2^n + 4 + 8*0^n).