A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-x*A(x))) * A(x).

1, 1, 4, 32, 393, 6547, 138046, 3525853, 105832964, 3651748332, 142429413387, 6196895235709, 297571887174040, 15632879134292045, 891910713837242092, 54919409605089141532, 3630105859259972654905, 256374187841461047791587

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..360

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+k+1)^(k-1) * Stirling2(n,k).

a(n) ~ sqrt((s-1)*s^3 / (1 + r*(2*s - 3)*s - r^2*(s-1)*s^2)) * n^(n-1) / (exp(n) * r^(n -1/2)), where r = 0.2202409288542107090687589144963703329896230236509... and s = 1.7315644042495989781932730410872588555151921253414... are roots of the system of equations s = s/exp(r*s) + log(s), (s-1)/s - (1 - r*s)/exp(r*s) = 0. - Vaclav Kotesovec, Nov 22 2021

Mathematica

a[n_] := Sum[(-1)^(n - k) * (n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(n+k+1)^(k-1)*stirling(n, k, 2));

Crossrefs

Cf. A058864, A349557.

Sequence in context: A195193 A377737 A203435 * A005263 A325574 A373027

Adjacent sequences: A349555 A349556 A349557 * A349559 A349560 A349561

Keyword

nonn

Author

Seiichi Manyama, Nov 21 2021

Status

approved