A349684 E.g.f. satisfies: log(A(x)) = 1 - exp(-x*A(x)^3).

1, 1, 6, 71, 1273, 30737, 935217, 34366971, 1481055674, 73255529901, 4090716385913, 254574063103175, 17471577758796377, 1310989371574276201, 106774436938943155714, 9381218495657924393523, 884444646528793096915853, 89063007333443317630241605

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..344

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (3*n+1)^(k-1) * Stirling2(n,k).

a(n) ~ s * n^(n-1) / (3*sqrt(1 - r*s^3) * exp(n) * r^n), where r = -LambertW(-1/3) / exp(3 + 1/LambertW(-1/3)) = 0.15501985846382288988548853891763630846... and s = exp(1 + 1/(3*LambertW(-1/3))) = 1.5865317583949486858973892879410781361... are roots of the system of equations exp(-r*s^3) + log(s) = 1, exp(r*s^3) = 3*r*s^3. - Vaclav Kotesovec, Nov 26 2021

Mathematica

nterms=20; Table[Sum[(-1)^(n-k)(3n+1)^(k-1)StirlingS2[n, k], {k, 0, n}], {n, 0, nterms-1}] (* Paolo Xausa, Nov 25 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(3*n+1)^(k-1)*stirling(n, k, 2));

Crossrefs

Cf. A349528, A349599.

Sequence in context: A092085 A028844 A274644 * A349598 A145089 A218676

Adjacent sequences: A349681 A349682 A349683 * A349685 A349686 A349687

Keyword

nonn

Author

Seiichi Manyama, Nov 25 2021

Status

approved