A099827 Generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5 multiplied by (n!)^5.

0, 1, 33, 8051, 8252000, 25795462624, 200610400564224, 3371852494046112768, 110492114540967125581824, 6524555433591956305924325376, 652461835742417609568446054400000, 105080260346474296336209157187174400000

Offset

0,3

Comments

Note that a(n) is divisible by n, except when n is prime. Also, a(n+1) is divisible by n, except when n is prime or n = 0.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..120

Eric Weisstein's World of Mathematics, Harmonic Number.

Formula

a(n) = (n!)^5 * Sum_{k=1..n} 1/k^5 = (n!)^5 * HarmonicNumber[n, 5] = (n!)^5 * A099828(n)/A069052(n).

a(0) = 0, a(1) = 1, a(n+1) = (n^5 + (n+1)^5)*a(n) - n^10*a(n-1) for n > 0. - Seiichi Manyama, Aug 24 2017

a(n) ~ Zeta(5) * (2*Pi)^(5/2) * n^(5*n+5/2) / exp(5*n). - Vaclav Kotesovec, Aug 27 2017

Sum_{n>=0} a(n) * x^n / (n!)^5 = polylog(5,x) / (1 - x). - Ilya Gutkovskiy, Jul 14 2020

Example

a(2) = (2!)^5 * (1 + 1/2^5) = 2^5 + 1 = 33,
a(3) = (3!)^5 * (1 + 1/2^5 + 1/3^5) = 6^5 + 3^5 + 1 = 8051.

Mathematica

Table[(n!)^5*Sum[1/k^5, {k, 1, n}], {n, 0, 13}] or Table[(n!)^5*HarmonicNumber[n, 5], {n, 0, 13}]

Crossrefs

Cf. A001008, A001819, A008515, A069052, A099828.

Column k = 5 of A291556.

Sequence in context: A219563 A183237 A099828 * A269793 A336197 A336261

Adjacent sequences: A099824 A099825 A099826 * A099828 A099829 A099830

Keyword

nonn

Author

Alexander Adamchuk, Oct 27 2004

Extensions

a(0) = 0 prepended by Seiichi Manyama, Aug 23 2017

Name edited by Petros Hadjicostas, May 10 2020

Status

approved