A291505 -id:A291505 - cp's OEIS frontend

A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 11, 4, 0, 1, 9, 49, 50, 5, 0, 1, 17, 251, 820, 274, 6, 0, 1, 33, 1393, 16280, 21076, 1764, 7, 0, 1, 65, 8051, 357904, 2048824, 773136, 13068, 8, 0, 1, 129, 47449, 8252000, 224021776, 444273984, 38402064, 109584, 9

Offset: 0

Seiichi Manyama, Aug 26 2017

			Square array begins:
   0,  0,   0,     0,      0, ...
   1,  1,   1,     1,      1, ...
   2,  3,   5,     9,     17, ...
   3, 11,  49,   251,   1393, ...
   4, 50, 820, 16280, 357904, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns k=0-10 give: A001477, A000254, A001819, A066989, A203229, A099827, A291456, A291505, A291506, A291507, A291508.
Rows n=0-3 give: A000004, A000012, A000051, A074528.
Main diagonal gives A060943.

A:= (n, k)-> n!^k * add(1/i^k, i=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 26 2017

A[0, ] = 0; A[1, ] = 1; A[n_, k_] := A[n, k] = ((n-1)^k + n^k) A[n-1, k] - (n-1)^(2k) A[n-2, k];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 11 2019 *)

A(0, k) = 0, A(1, k) = 1, A(n+1, k) = (n^k+(n+1)^k)*A(n, k) - n^(2*k)*A(n-1, k).

A291506 a(n) = (n!)^8 * Sum_{i=1..n} 1/i^8.

Original entry on oeis.org

0, 1, 257, 1686433, 110523752704, 43173450975314176, 72514862031522895036416, 418033821374598847702425993216, 7013444132843374500928464765799366656, 301905779820559925981495987360836056017534976

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..83

Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A099827 (k=5), A291456 (k=6), A291505 (k=7), this sequence (k=8), A291507 (k=9), A291508 (k=10).

Column k=8 of A291556.

Table[(n!)^8 * Sum[1/i^8, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) = n!^8*sum(i=1, n, 1/i^8); \\ Michel Marcus, Aug 26 2017

a(0) = 0, a(1) = 1, a(n+1) = (n^8+(n+1)^8)*a(n) - n^16*a(n-1) for n > 0.
a(n) ~ 8 * Pi^12 * n^(8*n+4) / (4725 * exp(8*n)). - Vaclav Kotesovec, Aug 27 2017
Sum_{n>=0} a(n) * x^n / (n!)^8 = polylog(8,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

A291507 a(n) = (n!)^9 * Sum_{i=1..n} 1/i^9.

Original entry on oeis.org

0, 1, 513, 10097891, 2647111616000, 5170142516807540224, 52103129720841632885243904, 2102549272223560776918400601161728, 282199388424234851655058321255905292713984, 109329825340451764123791003609208862665771818418176

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..75

Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A099827 (k=5), A291456 (k=6), A291505 (k=7), A291506 (k=8), this sequence (k=9), A291508 (k=10).

Column k=9 of A291556.

Table[(n!)^9 * Sum[1/i^9, {i, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) = n!^9*sum(i=1, n, 1/i^9); \\ Michel Marcus, Aug 26 2017

a(0) = 0, a(1) = 1, a(n+1) = (n^9+(n+1)^9)*a(n) - n^18*a(n-1) for n > 0.
a(n) ~ zeta(9) * (2*Pi)^(9/2) * n^(9*n+9/2) / exp(9*n). - Vaclav Kotesovec, Aug 27 2017
Sum_{n>=0} a(n) * x^n / (n!)^9 = polylog(9,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

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

Original entry on oeis.org

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

cp's OEIS Frontend

A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.

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A291506 a(n) = (n!)^8 * Sum_{i=1..n} 1/i^8.

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A291507 a(n) = (n!)^9 * Sum_{i=1..n} 1/i^9.

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

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