A086914 -id:A086914 - cp's OEIS frontend

A122778 a(n) = Sum_{k=0..n} A(n,k)*n^k where A(n,k) are Eulerian numbers.

1, 1, 3, 22, 285, 5656, 158095, 5881968, 279768825, 16507789696, 1180490926131, 100415158796800, 10005244013129365, 1152844128057793536, 151949197139815794615, 22696027820066041133056, 3810644613584486281328625

Offset: 0

Max Alekseyev, Sep 11 2006

nonn

Prime p divides a(p-1) for p>2. - Alexander Adamchuk, Sep 12 2006

Let A_n(x) denote the Eulerian polynomials with coefficients the Eulerian numbers as defined in the DLMF (number of permutations of {1,2,..,n} with k ascents) then a(n) = A_n(n). - Peter Luschny, Aug 09 2010

Digital Library of Mathematical Functions, Table 26.14.1
Eric Weisstein's World of Mathematics, Eulerian number at MathWorld
Eric Weisstein's World of Mathematics, Polylogarithm at MathWorld

Cf. A008292.

A122778 := n -> add(n^k*add((-1)^j*binomial(n+1,j)*(k-j+1)^n,j=0..k),k=0..n); # Peter Luschny, Aug 09 2010
seq(add(combinat:-eulerian1(n,k)*n^k,k=0..n),n=0..16); # Peter Luschny, Oct 19 2016

<< Combinatorica`; Table[Sum[Combinatorica`Eulerian[n, k] If[n == k == 0, 1, n^k], {k, 0, n}], {n, 0, 20}] (* Alexander Adamchuk, Sep 12 2006; corrected by Vladimir Reshetnikov, Oct 15 2016 *)
Flatten[{1, 1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n]/n, {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 16 2016 *)

a(n) = Sum_{k=0..n} A(n,k) * n^k
a(n) = Sum_{k=0..n} A(n,k) * n^(n-k).
a(n) = ((n-1)^(n+1))/n * Sum_{k>=1} k^n/n^k for n>1.
a(n) = ((n-1)^(n+1))/n * Li_{-n}(1/n) for n>1. - Alexander Adamchuk, Sep 12 2006
a(n) = (n-1)*A086914(n), n>1. - Vladeta Jovovic, Sep 12 2006
a(n) ~ exp(-1) * n! * n^n / log(n)^(n+1). - Vaclav Kotesovec, Jun 06 2022

a(0)=1 changed by Max Alekseyev, Nov 28 2011

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..271

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]

a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A122020 Sum[k=0..n] Eulerian[n,k]*n^k.

Original entry on oeis.org

1, 6, 66, 1140, 28280, 948570, 41173776, 2238150600, 148570107264, 11804909261310, 1104566746764800, 120062928157552380, 14986973664751315968, 2127288759957421124610, 340440417300990616995840

Offset: 1

Alexander Adamchuk, Sep 12 2006, Sep 14 2006

nonn

n divides a(n). 2^m divides a(n), where m(n) = {0,1,1,2,3,1,4,3,7,1,9,2,10,1,11,4,15,1,17,2,18,1,20,3,22,...}. p^k divides from a(p^k-1), a(p^k), a(p^k+1) for prime p>2 and integer k>0.

Eric Weisstein's World of Mathematics, Eulerian number
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A122778, A121376, A121985, A086914.

Table[Sum[Eulerian[n,k]*n^k,{k,0,n}],{n,1,25}]
Flatten[{1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n], {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 16 2016 *)

a(n) = Sum[ Eulerian[n,k]*n^(n-k-1), {k,0,n} ] = n*A122778[n]. a(n) = n(n-1)*A086914[n] for n>1. a(n) = ((n-1)^(n+1)) * PolyLog[ -n, 1/n ] = ((n-1)^(n+1)) * Sum[ k^n/n^k, {k,1,Infinity} ] = ((n-1)^(n+1)) * A121376[n]/A121985[n] for n>1.

a(n) ~ exp(-1) * n! * n^(n+1) / log(n)^(n+1). - Vaclav Kotesovec, Jun 06 2022

cp's OEIS Frontend

A122778 a(n) = Sum_{k=0..n} A(n,k)*n^k where A(n,k) are Eulerian numbers.

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A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

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A122020 Sum[k=0..n] Eulerian[n,k]*n^k.

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