A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.

1, 1, -2, 17, -213, 3712, -82773, 2250565, -72218912, 2671680015, -111950278213, 5240764049094, -271082407059027, 15353947287972373, -945097225235334538, 62820021683240176445, -4484426869618973019249, 342169496779859317566456

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..356

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

Formula

a(n) = Sum_{k=0..n} (-2*k+1)^(k-1) * Stirling2(n,k).

E.g.f.: A(x) = exp( LambertW(2*(exp(x) - 1))/2 ).

G.f.: Sum_{k>=0} (-2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).

a(n) ~ -(-1)^n * sqrt(2*exp(1) - 1) * sqrt(log(2) - log(2 - exp(-1))) * n^(n-1) / (2 * exp(n + 1/2) * (log(2) - log(2*exp(1) - 1) + 1)^n). - Vaclav Kotesovec, Nov 24 2021

Maple

b:= proc(n, m) option remember; `if`(n=0,
     (1-2*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[(-2*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, (-2*k+1)^(k-1)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1))/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

Crossrefs

Cf. A349650, A349652, A349656.

Cf. A008277, A349583, A349655.

Sequence in context: A364335 A364446 A333990 * A364333 A370289 A004029

Adjacent sequences: A349651 A349652 A349653 * A349655 A349656 A349657

Keyword

sign

Author

Seiichi Manyama, Nov 23 2021

Status

approved