A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3*k+1)^(k-1) * Stirling2(n,k).

1, 1, 6, 80, 1645, 45962, 1627080, 69817575, 3522349232, 204343964292, 13403304111515, 980876342339456, 79235384391436316, 7003257362607771709, 672285536392973397658, 69656231091367157111844, 7747832754070176901631621

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..336

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

Formula

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

E.g.f.: 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) ~ sqrt(3*exp(1) - 1) * sqrt(log(3/(3-exp(-1)))) * n^(n-1) / (3 * exp(n - 1/3) * (1 + log(3/(3*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

Mathematica

a[n_] := Sum[(-1)^(n - k)*(3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 21 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(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. A008277, A058864, A196556, A349525, A349527.

Sequence in context: A167570 A337564 A177776 * A132616 A349657 A323694

Adjacent sequences: A349525 A349526 A349527 * A349529 A349530 A349531

Keyword

nonn

Author

Seiichi Manyama, Nov 20 2021

Status

approved