A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.

1, 1, -4, 50, -981, 26632, -924526, 39114343, -1952373450, 112321286934, -7318049389727, 532602419776770, -42825957593127770, 3770431528821292441, -360734325565272740984, 37267364164988863692782, -4134667018838875759388749, 490302545213321842575157140

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..337

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

Formula

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

E.g.f.: A(x) = exp( LambertW(3*(exp(x) - 1))/3 ).

G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).

a(n) ~ -(-1)^n * sqrt(3*exp(1) - 1) * sqrt(log(3) - log(3 - exp(-1))) * n^(n-1) / (3 * exp(n + 1/3) * (log(3) - log(3*exp(1) - 1) + 1)^n). - Vaclav Kotesovec, Nov 24 2021

Maple

b:= proc(n, m) option remember; `if`(n=0,
     (1-3*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022

Mathematica

a[n_] := Sum[(-3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-3*k+1)^(k-1)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(exp(x)-1))/3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-3*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

Crossrefs

Cf. A349651, A349653, A349657.

Cf. A008277, A349583, A349654.

Sequence in context: A123356 A234870 A139087 * A173218 A234714 A201829

Adjacent sequences: A349652 A349653 A349654 * A349656 A349657 A349658

Keyword

sign

Author

Seiichi Manyama, Nov 23 2021

Status

approved