A066989 -id:A066989 - cp's OEIS frontend

A291508 a(n) = (n!)^10 * Sum_{i=1..n} 1/i^10.

0, 1, 1025, 60526249, 63466432537600, 619789443653380965376, 37476298202061058687475122176, 10586126703664512292193022557971021824, 11366767006463449393869821987386636472445566976, 39633465899293694663690352980684333029782095493517541376

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..69

Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A099827 (k=5), A291456 (k=6), A291505 (k=7), A291506 (k=8), A291507 (k=9), this sequence (k=10).

Column k=10 of A291556.

Table[(n!)^10 * Sum[1/i^10, {i, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) = n!^10*sum(i=1, n, 1/i^10); \\ Michel Marcus, Aug 26 2017

a(0) = 0, a(1) = 1, a(n+1) = (n^10+(n+1)^10)*a(n) - n^20*a(n-1) for n > 0.
a(n) ~ 32 * Pi^15 * n^(10*n+5) / (93555 * exp(10*n)). - Vaclav Kotesovec, Aug 27 2017
Sum_{n>=0} a(n) * x^n / (n!)^10 = polylog(10,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

A291585 a(n) = ((2n-1)!!)^3 * Sum_{i=1..n} 1/(2*i-1)^3.

Original entry on oeis.org

0, 1, 28, 3527, 1213136, 885533769, 1179489355164, 2592461355675183, 8752024838912328000, 43007026735417743104625, 295026115290377469777013500, 2732517516522603185416699320375, 33249139841891876848539850177362000, 519549434719363290227226596791835765625

Offset: 0

Seiichi Manyama, Aug 27 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..160

Cf. A001147, A066989.

Table[(2*n-1)!!^3 * Sum[1/(2*i-1)^3, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = ((2*n-1)^3+(2*n+1)^3)*a(n) - (2*n-1)^6*a(n-1) for n > 0.
a(n) ~ 7*Zeta(3) * 2^(3*n-3/2) * n^(3*n) / exp(3*n). - Vaclav Kotesovec, Aug 27 2017
a(1) = 1, a(n) = a(n-1)*(2n-1)^3 + ((2n-1)!!)^3/(2n-1)^3, for n > 1. - Daniel Suteu, Nov 01 2017

A203156 (n-1)-st elementary symmetric function of {4,9,16,25,..., (n+1)^2}.

Original entry on oeis.org

1, 13, 244, 6676, 254736, 13000464, 857431296, 71077637376, 7239445632000, 889141110912000, 129629670893568000, 22136856913815552000, 4377599743151480832000, 992559996665635184640000, 255805371399126806691840000

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

			Let esf abbreviate "elementary symmetric function". Then
0th esf of {4}:  1;
1st esf of {4,9}:  4 + 9 = 13;
2nd esf of {4,9,16}:  4*9 + 4*16 + 9*16 = 244.

Robert Israel, Table of n, a(n) for n = 1..251

Cf. A066989.

f:= proc(n) local i,x; coeff(mul(x+i^2,i=2..n+1),x,1) end proc:
map(f, [$1..26]); # Robert Israel, Dec 16 2024

f[k_] := (k + 1)^2; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A203156 *)

a(n) = gamma(2 + n)^2*(Pi^2/6 - 1 - digamma^(1)(2 + n)), where gamma(x) is the gamma function and digamma^(n)(x) is the n-th derivative of the digamma function. - Markus Bindhammer, Nov 26 2017

cp's OEIS Frontend

A291508 a(n) = (n!)^10 * Sum_{i=1..n} 1/i^10.

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A291585 a(n) = ((2n-1)!!)^3 * Sum_{i=1..n} 1/(2*i-1)^3.

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A203156 (n-1)-st elementary symmetric function of {4,9,16,25,..., (n+1)^2}.

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