A213132 -id:A213132 - cp's OEIS frontend

A213131 Polylogarithm li(-n,-1/8) multiplied by (9^(n+1))/8.

1, -1, -7, -33, 105, 5199, 64953, -46593, -21769335, -497664081, -1941272487, 256114020447, 9566995408425, 99966666676239, -6245895772363527, -366865939437422913, -6924777575908002615, 259022993102904450159, 24387711970312991335833, 716398360186298080983327

Offset: 0

Stanislav Sykora, Jun 06 2012

sign

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=8.

			polylog(-5,-1/8)*9^6/8 = 5199.

Stanislav Sykora, Table of n, a(n) for n = 0..100
OEIS-Wiki, Eulerian polynomials

Cf. A212846, A210246, A212847, A213127 to A213130

Cf. A213132 to A213157

seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*8^k, k=0..n),n=0..17); # Peter Luschny, Apr 21 2013

Table[If[n == 0, 1, PolyLog[-n, -1/8] 9^(n+1)/8], {n, 0, 19}] (* Jean-François Alcover, Jun 27 2019 *)

/* See A212846; run limnpq(nmax, 1, 8) */

x='x+O('x^66); Vec(serlaplace( 9/(8+exp(9*x)) )) \\ Joerg Arndt, Apr 21 2013

a(n) = sum(k=0, n, k!*(-1)^k*9^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

See formula in A212846, setting p=1,q=8.
E.g.f.: 9/(8+exp(9*x)). [Joerg Arndt, Apr 21 2013]
a(n) = Sum_{k=0..n} k! * (-1)^k * 9^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

A213133 Polylogarithm li(-n,-1/10) multiplied by (11^(n+1))/10.

Original entry on oeis.org

1, -1, -9, -61, -9, 9659, 197631, 1388099, -51302169, -2339721781, -41290278129, 536297904659, 64956862241271, 2152254297009179, 6320179650231711, -3288155212484644381, -187761119883430045689

Offset: 0

Stanislav Sykora, Jun 06 2012

sign

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=10.

			polylog(-5,-1/10)*11^6/10 = 9659.

Stanislav Sykora, Table of n, a(n) for n = 0..100

Cf. A212846, A210246, A212847, A213127 to A213132.

Cf. A213134 to A213157.

f[n_] := PolyLog[-n, -1/10] 11^(n + 1)/10; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Dec 25 2015 *)

PARI
```
\\ in A212846; run limnpq(nmax, 1, 10)
```

a(n) = sum(k=0, n, k!*(-1)^k*11^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

See formula in A212846, setting p=1,q=10.

a(n) = Sum_{k=0..n} k! * (-1)^k * 11^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

A355373 a(n) = Sum_{k=0..n} k! * (-1)^k * n^(n-k) * Stirling2(n,k).

Original entry on oeis.org

1, -1, 0, 3, 40, 455, 2016, -177373, -11564160, -497664081, -12796467200, 536297904659, 132025634657280, 14907422733429239, 1181852660381503488, 34684559693802943875, -11771644802057621110784, -3553614228958108389522721, -656899368126170250221715456

Offset: 0

Seiichi Manyama, Jun 30 2022

sign

Cf. A212846, A213127, A213128, A213129, A213130, A213131, A213132, A213133.

Cf. A318183, A352074.

a[n_] := Sum[k! * (-1)^k * n^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)

a(n) = sum(k=0, n, k!*(-1)^k*n^(n-k)*stirling(n, k, 2));

a(n) = n! * [x^n] n/(n - 1 + exp(n*x)) for n > 0.

cp's OEIS Frontend

A213131 Polylogarithm li(-n,-1/8) multiplied by (9^(n+1))/8.

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A213133 Polylogarithm li(-n,-1/10) multiplied by (11^(n+1))/10.

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A355373 a(n) = Sum_{k=0..n} k! * (-1)^k * n^(n-k) * Stirling2(n,k).

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