A121376 -id:A121376 - cp's OEIS frontend

A121985 Denominator of PolyLog(-n, 1/n).

12, 1, 8, 81, 512, 15625, 11664, 5764801, 8388608, 387420489, 78125000, 3138428376721, 5159780352, 3937376385699289, 21703138331168, 1081219482421875, 144115188075855872, 14063084452067724991009

Offset: 1

Alexander Adamchuk, Sep 10 2006, Sep 14 2006

PolyLog(n,z) = Sum_{k>=1} z^k/k^n. PolyLog(-n, 1/n) = Sum_{k>=1} k^n/n^k for n > 1. Numerators of PolyLog(-n, 1/n) are listed in A121376(n) = {-1, 6, 33, 380, 3535, 189714, 285929, ...}. a(p+1) = p^(p+1) for prime p. a(p^k+1) = p^( k*p^k + 2*k - (p^k - 1)/(p - 1) ) for prime p and integer k > 0. Prime divisors of a(n) are the same as prime divisors of (n-1).
It appears that for most squarefree (n-1) if q is the largest prime divisor of (n-1) then q^(n - (n-1)/q + 1) divides a(n).
PolyLog(-n, 1/n) = A121376(n) / A121985(n) = (Sum_{k=0..n} Eulerian(n,k) * n^(n-k+1)) / (n-1)^(n+1) = n*A122778(n) = (Sum_{k=0..n} Eulerian(n,k) * n^k) / (n-1)^(n+1) = A122020(n) for n > 1.

			PolyLog(-n, 1/n) begins -1/12, 6, 33/8, 380/81, 3535/512, 189714/15625, ...
a(3) = 2^3;
a(4) = 3^4;
a(200) = 199^200;
a(257) = 2^1809;
a(290) = 17^564;
a(319) = 2^7 * 3^164 * 53^314, where 2*3*53 = 318 = 319 - 1 and 314 = 319 - 319/53 + 1;
a(709) = 2^716 * 3^360 * 59^698;
a(710) = 709^710.

G. C. Greubel, Table of n, a(n) for n = 1..430
Eric Weisstein's World of Mathematics, Polylogarithm.
Eric Weisstein's World of Mathematics, Eulerian number

Cf. A121376 = numerators.

Cf. A120020, A122778.

Join[{12}, Table[Denominator[PolyLog[ -n,1/n]],{n,2,30}]]

a(n)=denominator(polylog(-n,1/n)) \\ Charles R Greathouse IV, Jul 14 2014

a(n) = denominator(PolyLog(-n, 1/n)).

a(n) = denominator((-1)^(n+1) * PolyLog(-n, n)).

A086914 a(n) = ((n-1)^n/n)*Sum_{k>=1} (k^n/n^k).

Original entry on oeis.org

0, 3, 11, 95, 1414, 31619, 980328, 39966975, 2063473712, 131165658459, 10041515879680, 909567637557215, 96070344004816128, 11688399779985830355, 1621144844290431509504, 254042974238965752088575

Offset: 1

Benoit Cloitre, Sep 24 2003

nonn

Appears to always be an integer.

Cf. A122020, A122778, A121376, A121985.

Table[Sum[(n-1)^n*k^n/n^(k+1), {k, 1, Infinity}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 16 2016 *)

a(n) = Euler(n, n)/(n-1) where Euler(n, x) is Eulerian polynomial of degree n (cf. A008292). - Vladeta Jovovic, Sep 26 2003
a(n) = (n-1)^n/n*polylog(-n, 1/n) = 1/(n-1)*Sum(n^i*Sum((-1)^j*binomial(n+1, j)*(i-j+1)^n, j = 0 .. i), i = 0 .. n), n>1. - Vladeta Jovovic, Sep 26 2003
Prime p divides a(p-1) for p>2. - Alexander Adamchuk, Sep 19 2006
a(n) = A122020[n] / (n*(n-1)) for n>1. a(n) = A122778[n] / (n-1) for n>1. a(n) = ((n-1)^n)/n * A121376[n]/A121985[n] for n>1. - Alexander Adamchuk, Sep 19 2006
a(n) ~ exp(-1) * n! * n^(n-1) / log(n)^(n+1). - Vaclav Kotesovec, Jun 06 2022

A122020 Sum[k=0..n] Eulerian[n,k]*n^k.

Original entry on oeis.org

1, 6, 66, 1140, 28280, 948570, 41173776, 2238150600, 148570107264, 11804909261310, 1104566746764800, 120062928157552380, 14986973664751315968, 2127288759957421124610, 340440417300990616995840

Offset: 1

Alexander Adamchuk, Sep 12 2006, Sep 14 2006

nonn

n divides a(n). 2^m divides a(n), where m(n) = {0,1,1,2,3,1,4,3,7,1,9,2,10,1,11,4,15,1,17,2,18,1,20,3,22,...}. p^k divides from a(p^k-1), a(p^k), a(p^k+1) for prime p>2 and integer k>0.

Eric Weisstein's World of Mathematics, Eulerian number
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A122778, A121376, A121985, A086914.

Table[Sum[Eulerian[n,k]*n^k,{k,0,n}],{n,1,25}]
Flatten[{1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n], {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 16 2016 *)

a(n) = Sum[ Eulerian[n,k]*n^(n-k-1), {k,0,n} ] = n*A122778[n]. a(n) = n(n-1)*A086914[n] for n>1. a(n) = ((n-1)^(n+1)) * PolyLog[ -n, 1/n ] = ((n-1)^(n+1)) * Sum[ k^n/n^k, {k,1,Infinity} ] = ((n-1)^(n+1)) * A121376[n]/A121985[n] for n>1.

a(n) ~ exp(-1) * n! * n^(n+1) / log(n)^(n+1). - Vaclav Kotesovec, Jun 06 2022

A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 3, 26, 1, 5, 20, 33, 150, 1, 6, 15, 44, 15, 1082, 1, 7, 42, 115, 380, 273, 9366, 1, 8, 7, 366, 285, 4108, 1491, 94586, 1, 9, 72, 91, 4074, 3535, 17780, 38001, 1091670, 1, 10, 45, 776, 70, 11334, 26355, 269348, 17295, 14174522, 1, 11, 110, 531, 10440, 2149, 189714, 458555, 4663060, 566733, 204495126

Offset: 0

Mohammed Yaseen, Jul 22 2024

			Array begins:
+-----+--------------------------------------------------------------+
| n\k |       2     3       4       5        6      7         8  ... |
+-----+--------------------------------------------------------------+
|  0  |       1     1       1       1        1      1         1  ... |
|  1  |       2     3       4       5        6      7         8  ... |
|  2  |       6     3      20      15       42      7        72  ... |
|  3  |      26    33      44     115      366     91       776  ... |
|  4  |     150    15     380     285     4074     70     10440  ... |
|  5  |    1082   273    4108    3535    11334   2149    174728  ... |
|  6  |    9366  1491   17780   26355   189714   3311   3525192  ... |
|  7  |   94586 38001  269348  458555  3706518 285929  11870648  ... |
|  8  | 1091670 17295 4663060 1139685 82749954 220430 319735800  ... |
| ... |     ...   ...     ...     ...      ...    ...       ...  ... |
+-----+--------------------------------------------------------------+

Cf. A374896 (denominators).

Cf. A000629, A008292, A121376, A276805.

T(n,k) = numerator(polylog(-n, 1/k));
matrix(7,7,n,k,T(n-1, k+1)) \\ Michel Marcus, Aug 04 2024

T(n,k) = numerator(polylog(-n, 1/k)).
T(n,k) = numerator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
T(0,k) = 1.
T(1,k) = k.
T(2,k) = A276805(k-1).
T(n,2) = A000629(n).
T(n,n) = A121376(n).

cp's OEIS Frontend

A121985 Denominator of PolyLog(-n, 1/n).

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A086914 a(n) = ((n-1)^n/n)*Sum_{k>=1} (k^n/n^k).

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A122020 Sum[k=0..n] Eulerian[n,k]*n^k.

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A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

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