A351136 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(2*n) * Stirling1(n,k).

1, 1, 33, 4760, 1814698, 1436035954, 2041681617638, 4736066140912728, 16729538152432476024, 85437808930634601070944, 605822464949212598847700512, 5774077466357788471179323050704, 72030066703292325305595937373723040

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..160

Formula

E.g.f.: Sum_{k>=0} (-log(1 - k^2*x))^k.

a(n) ~ c * r^(2*n) * (1 + r*exp(2/r))^n * n^(3*n + 1/2) / exp(3*n), where r = 0.9414380538633895499299457441124149470954491698433... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-2/r) and c = 2.22047212763474863127102273073825610210704559048894... - Vaclav Kotesovec, Feb 03 2022

Mathematica

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(2*n)*stirling(n, k, 1));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^2*x))^k)))

Crossrefs

Cf. A007840, A320096, A351137.

Cf. A187755, A351133, A351138.

Sequence in context: A350722 A354674 A229260 * A232365 A183551 A114071

Adjacent sequences: A351133 A351134 A351135 * A351137 A351138 A351139

Keyword

nonn

Author

Seiichi Manyama, Feb 02 2022

Status

approved