A351181 -id:A351181 - cp's OEIS frontend

A351180 a(n) = Sum_{k=0..n} k^(k+n) * Stirling1(n,k).

1, 1, 15, 635, 53112, 7367444, 1529130770, 443685287576, 171495189203456, 85174828026304824, 52856314387144232184, 40077340463437963801752, 36457068309928364981668848, 39186634107857517367884040632

Offset: 0

Seiichi Manyama, Feb 04 2022

nonn

Cf. A305819, A320082, A350726, A351182, A351183.

Cf. A350721, A351181, A351769.

a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)

a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+k*x))^k/k!)))

E.g.f.: Sum_{k>=0} (k * log(1 + k*x))^k / k!.

a(n) ~ c * d^n * n^(2*n), where d = 0.9315722818790917570256960813246568629715677803436281996798798428122211769... and c = 1.07238575181275524934156216072811545518508724720534339814911465361... - Vaclav Kotesovec, Feb 18 2022

A351769 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * k^(k+n).

Original entry on oeis.org

1, 1, 17, 827, 79368, 12623124, 3002832110, 998401869464, 442148442609152, 251578963946182968, 178846127724854653704, 155339277405600252114072, 161863497852092601156187152, 199286757107586767535516731832, 286210094619439661737214469710088

Offset: 0

Vaclav Kotesovec, Feb 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..213

Cf. A351180, A351181.

Table[Sum[k^(k+n) * StirlingS1[n, k] * (-1)^(n-k), {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, (-1)^(n-k)*stirling(n, k, 1)*k^(k+n)); \\ Michel Marcus, Feb 19 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k*log(1-k*x))^k/k!))) \\ Seiichi Manyama, Jun 02 2022

a(n) ~ c * r^n * (1 + r*exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.937997555632908331545534056235449048849427140626270261830822459734975609... is the root of the equation r + exp(-1 - 1/r) = -LambertW(-1, -r*exp(-r)) and c = 0.9367460233410089838603007174937882495902299959682250862650092226619624... - Vaclav Kotesovec, Feb 18 2022

E.g.f.: Sum_{k>=0} (-k * log(1 - k*x))^k / k!. - Seiichi Manyama, Jun 02 2022

cp's OEIS Frontend

A351180 a(n) = Sum_{k=0..n} k^(k+n) * Stirling1(n,k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A351769 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * k^(k+n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula