A291556 -id:A291556 - cp's OEIS frontend

A066989 a(n) = (n!)^3 * Sum_{i=1..n} 1/i^3.

1, 9, 251, 16280, 2048824, 444273984, 152759224512, 78340747014144, 57175952894078976, 57223737619918848000, 76212579497951858688000, 131758938842553681444864000, 289584291977410916858462208000, 794860754824699647616459210752000

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

p^2 divides a(p-1) for prime p>5. - Alexander Adamchuk, Jul 11 2006

Seiichi Manyama, Table of n, a(n) for n = 1..181 (terms 1..50 from T. D. Noe)

Cf. A007408.
Cf. A001819, A143003, A143004, A143005, A143006.
Column k=3 of A291556.

f[k_] := k^3; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A066989 *)
(* Clark Kimberling, Dec 29 2011 *)
Table[(n!)^3 * Sum[1/i^3, {i, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 27 2017 *)

Recurrence: a(1) = 1, a(2) = 9, a(n+2) = (2*n+3)*(n^2+3*n+3)*a(n+1) - (n+1)^6*a(n). b(n) = n!^3 satisfies the same recurrence with the initial conditions b(1) = 1, b(2) = 8. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(1-1^6/(9-2^6/(35-3^6/(91-...-(n-1)^6/((2n-1)*(n^2-n+1)))))) for n >= 2, leading to the infinite continued fraction expansion zeta(3) = 1/(1-1^6/(9-2^6/(35-3^6/(91-...-(n-1)^6/((2n-1)*(n^2-n+1)-...))))). Compare with A001819. - Peter Bala, Jul 19 2008
a(n) ~ Zeta(3) * (2*Pi)^(3/2) * n^(3*n+3/2) / exp(3*n). - Vaclav Kotesovec, Aug 27 2017
Sum_{n>=1} a(n) * x^n / (n!)^3 = polylog(3,x) / (1 - x). - Ilya Gutkovskiy, Jul 14 2020

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

Original entry on oeis.org

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

Offset: 0

Alexander Adamchuk, Oct 27 2004

nonn

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.

			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.

Seiichi Manyama, Table of n, a(n) for n = 0..120
Eric Weisstein's World of Mathematics, Harmonic Number.

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

Column k = 5 of A291556.

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

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

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

Name edited by Petros Hadjicostas, May 10 2020

A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

Original entry on oeis.org

0, 1, 65, 47449, 194397760, 3037656102976, 141727869124448256, 16674281388691716870144, 4371079210518164503303028736, 2322975003299339366419974718488576, 2322977286679362958150790503464960000000

Offset: 0

Seiichi Manyama, Aug 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..104

Column k=6 of A291556.
Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A203229 (k=4), A099827 (k=5).
Cf. A008516, A103345, A103346.

Table[(n!)^6 * Sum[1/i^6, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = (n^6 + (n+1)^6)*a(n) - n^12*a(n-1) for n > 0.
a(n) ~ 8 * Pi^9 * n^(6*n+3) / (945 * exp(6*n)). - Vaclav Kotesovec, Aug 27 2017
a(n) = (n!)^6 * A103345(n)/A103346(n). - Petros Hadjicostas, May 10 2020
Sum_{n>=0} a(n) * x^n / (n!)^6 = polylog(6,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

A291505 a(n) = (n!)^7 * Sum_{i=1..n} 1/i^7.

Original entry on oeis.org

0, 1, 129, 282251, 4624680320, 361307736471424, 101143400834944548864, 83296040059942781485105152, 174684539610200377980575079727104, 835510910973061065615656036610946891776, 8355109938323553617123838798161699143680000000

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..92

Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A099827 (k=5), A291456 (k=6), this sequence (k=7), A291506 (k=8), A291507 (k=9), A291508 (k=10).

Column k=7 of A291556.

Table[(n!)^7 * Sum[1/i^7, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) = n!^7*sum(i=1, n, 1/i^7); \\ Michel Marcus, Aug 26 2017

a(0) = 0, a(1) = 1, a(n+1) = (n^7+(n+1)^7)*a(n) - n^14*a(n-1) for n > 0.
a(n) ~ zeta(7) * (2*Pi)^(7/2) * n^(7*n+7/2) / exp(7*n). - Vaclav Kotesovec, Aug 27 2017
Sum_{n>=0} a(n) * x^n / (n!)^7 = polylog(7,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

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

A060943 a(n) = n!^n * Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 5, 251, 357904, 25795462624, 141727869124448256, 83296040059942781485105152, 7013444132843374500928464765799366656, 109329825340451764123791003609208862665771818418176, 396334659032531033249146049131230887376087800711479296000000000000

Offset: 1

Leroy Quet, May 07 2001

			a(3) = 6^3 *(1 + 1/2^3 + 1/3^3) = 251.

Harry J. Smith, Table of n, a(n) for n = 1..30

Cf. A036740.

Main diagonal of A291556.

[(Factorial(n))^n*(&+[1/j^n: j in [1..n]]): n in [1..15]]; // G. C. Greubel, Apr 09 2021

A060943:= n-> (n!)^n*add(1/j^n, j=1..n); seq(A060943(n), n=1..15); # G. C. Greubel, Apr 09 2021

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

{ default(realprecision, 100); for (n=1, 30, write("b060943.txt", n, " ", n!^n * sum(k=1, n, 1/k^n)); ) } \\ Harry J. Smith, Jul 14 2009

[(factorial(n))^n*sum(1/j^n for j in (1..n)) for n in (1..15)] # G. C. Greubel, Apr 09 2021

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Aug 27 2017

a(n) = (n!)^n * [x^n] PolyLog(n,x)/(1 - x), where PolyLog() is the polylogarithm function. - Ilya Gutkovskiy, Nov 27 2017

A203229 (n-1)-st elementary symmetric function of (1,16,...,n^4).

Original entry on oeis.org

1, 17, 1393, 357904, 224021776, 290539581696, 697854274212096, 2859056348455305216, 18760911610508623282176, 187626456226399005573120000, 2747212346823835568109649920000, 56968733990900457398848318341120000

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A000583, A203148.

Column k=4 of A291556.

f[k_] := k^4; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 14}]     (* A203229 *)
Table[(n!)^4 * Sum[1/i^4, {i, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) ~ 2 * Pi^6 * n^(4*n+2) / (45*exp(4*n)). - Vaclav Kotesovec, Aug 27 2017

Sum_{n>=1} a(n) * x^n / (n!)^4 = polylog(4,x) / (1 - x). - Ilya Gutkovskiy, Jul 14 2020

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 10, 23, 4, 0, 1, 28, 259, 176, 5, 0, 1, 82, 3527, 12916, 1689, 6, 0, 1, 244, 51331, 1213136, 1057221, 19524, 7, 0, 1, 730, 762743, 123296356, 885533769, 128816766, 264207, 8, 0, 1, 2188, 11406979, 12820180976, 809068942341, 1179489355164, 21878089479, 4098240, 9

Offset: 0

Seiichi Manyama, Aug 28 2017

			Square array begins:
  0,   0,     0,       0,         0, ...
  1,   1,     1,       1,         1, ...
  2,   4,    10,      28,        82, ...
  3,  23,   259,    3527,     51331, ...
  4, 176, 12916, 1213136, 123296356, ...

Seiichi Manyama, Antidiagonals n = 0..54, flattened

Columns k=0-5 give: A001477, A004041(n+1), A001824(n+1), A291585, A291586, A291587.
Rows n=0-2 give: A000004, A000012, A034472.
Main diagonal gives A291676.
Cf. A291556.

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

cp's OEIS Frontend

A066989 a(n) = (n!)^3 * Sum_{i=1..n} 1/i^3.

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

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A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

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A291505 a(n) = (n!)^7 * Sum_{i=1..n} 1/i^7.

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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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A060943 a(n) = n!^n * Sum_{k=1..n} 1/k^n.

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