A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1)^(k-1) * Stirling2(n,k).

1, 1, 4, 35, 469, 8502, 194807, 5402497, 175985390, 6587650497, 278674144201, 13148017697608, 684554867667117, 38988819551585477, 2411411875573335044, 160951864352781351959, 11531509389384310870257

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

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

E.g.f.: 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) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2-exp(-1)))) * n^(n-1) / (2 * exp(n - 1/2) * (1 + log(2/(2*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

Mathematica

a[n_] := Sum[(-1)^(n - k) * (2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(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. A008277, A058864, A196555, A349524, A349528.

Sequence in context: A367925 A367923 A242795 * A287886 A342207 A224797

Adjacent sequences: A349524 A349525 A349526 * A349528 A349529 A349530

Keyword

nonn

Author

Seiichi Manyama, Nov 20 2021

Status

approved