A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..399

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

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

E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).

G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).

a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

Maple

b:= proc(n, m) option remember; `if`(n=0,
     (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

Mathematica

a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-k+1)^(k-1)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(exp(x)-1))))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

Crossrefs

Cf. A120980, A349561, A349585.

Cf. A008277, A052880, A349524, A349525.

Sequence in context: A121879 A118789 A258114 * A370889 A367485 A133941

Adjacent sequences: A349580 A349581 A349582 * A349584 A349585 A349586

Keyword

sign

Author

Seiichi Manyama, Nov 22 2021

Status

approved