A080163 -id:A080163 - cp's OEIS frontend

A227044 a(n) = Sum_{k>=1} k^(2*n)/(2^k).

1, 6, 150, 9366, 1091670, 204495126, 56183135190, 21282685940886, 10631309363962710, 6771069326513690646, 5355375592488768406230, 5149688839606380769088406, 5916558242148290945301297750, 8004451519688336984972255078166, 12595124129900132067036747870669270

Offset: 0

Vaclav Kotesovec, Jun 29 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Bisection of A000629.

Cf. A080163.

Table[Sum[k^(2*n)/(2^k), {k, 1, Infinity}], {n, 0, 20}]
a[n_] := PolyLog[-2 n, 1/2]; a[0] = 1; Array[a, 15, 0] (* Peter Luschny, Sep 06 2020 *)

{a(n) = sum(k=0, 2*n, (-2)^k * k! * stirling(2*n, k,2) )}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ (2n)!/(log(2))^(2*n+1).
a(n) = Sum_{k=0..2*n} (-2)^k * k! * Stirling2(2*n, k). - Paul D. Hanna, Apr 15 2018
a(n) = A000629(2*n). - Christian Krause, Nov 22 2022

A340837 a(n) = (1/2) * Sum_{k>=0} (k*(k - 1))^n / 2^k.

Original entry on oeis.org

1, 2, 52, 3272, 382672, 71819552, 19755648832, 7489898916992, 3743721038908672, 2385494267756237312, 1887436919680269939712, 1815491288416066631616512, 2086364959404184854563049472, 2823211429546048668686123343872, 4443155724532239407325655263035392

Offset: 0

Ilya Gutkovskiy, Jan 23 2021

nonn

Cf. A000670, A020556, A052841, A080163, A098696, A122101, A249938.

Table[(1/2) Sum[(k (k - 1))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Table[(1/2) Sum[(-1)^k Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0], {k, 0, n}], {n, 0, 14}]

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A000670(2*n-k).
a(n) = 2 * A080163(n) for n > 0. - Hugo Pfoertner, Jan 23 2021
a(n) = A122101(2*n,n). - Alois P. Heinz, Jun 23 2023

cp's OEIS Frontend

A227044 a(n) = Sum_{k>=1} k^(2*n)/(2^k).

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