A349656 E.g.f. satisfies: A(x)^2 * log(A(x)) = 1 - exp(-x).

1, 1, -4, 35, -515, 10662, -284105, 9255185, -356346618, 15831168657, -797090201295, 44853942667096, -2789671436309939, 190023794141566309, -14069208182313480292, 1124994237749880216439, -96618656489949875115879, 8870165918232448251272870

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..347

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

Formula

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

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

Mathematica

a[n_] := (-1)^(n - 1) * 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) = (-1)^(n-1)*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*(1-exp(-x)))/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, A349654.

Cf. A008277, A349585, A349657.

Sequence in context: A342207 A224797 A143669 * A180716 A171778 A119391

Adjacent sequences: A349653 A349654 A349655 * A349657 A349658 A349659

Keyword

sign

Author

Seiichi Manyama, Nov 23 2021

Status

approved