A121985 -id:A121985 - cp's OEIS frontend

A121376 Numerator of PolyLog(-n, 1/n).

-1, 6, 33, 380, 3535, 189714, 285929, 319735800, 1160703963, 145739620510, 86294277091, 10914811650686580, 60229285882649, 163637596919801624970, 3392462704290802545, 669084376596453009616, 370468452361579892135179, 157145213515550643044429571846

Offset: 1

Alexander Adamchuk, Sep 06 2006

			PolyLog(-n, 1/n) begins -1/12, 6, 33/8, 380/81, 3535/512, 189714/15625, ...

Thomas Scheuerle, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A121985 (denominator).

Cf. A119758.

Numerator[Table[PolyLog[ -n,1/n],{n,1,40}]]

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

a(n) = numerator( PolyLog( -n, 1/n ) ). For n>1 a(n) = numerator( (-1)^n * PolyLog( -n, n ) ).
PolyLog(-n, 1/n) = a(n)/A121985(n) = Sum_{k>=1} k^n/n^k, for n > 1. n divides a(n). p^k divides a(p^k) for all prime p and integer k>0. p^k divides a(p^k-1) for prime p>2 and integer k>0. Also PolyLog(n, z) = Sum_{k>=1} z^k/k^n.
For n>1, a(n) is the numerator of n*A122778(n)/(n-1)^(n+1) = Sum_{k=0..n} A(n,k)*n^(k+1)/(n-1)^(n+1). For n>1, a(n) = n * A122778(n)/gcd(A122778(n),(n-1)^(n+1)). - Max Alekseyev, Sep 11 2006

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

A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 2, 1, 5, 16, 27, 8, 1, 6, 25, 32, 27, 1, 1, 7, 36, 125, 128, 81, 4, 1, 8, 49, 27, 625, 128, 243, 4, 1, 9, 64, 343, 216, 3125, 512, 243, 16, 1, 10, 81, 256, 2401, 81, 3125, 1024, 729, 1, 1, 11, 100, 729, 2048, 16807, 972, 15625, 4096, 2187, 4, 1

Offset: 0

Mohammed Yaseen, Aug 03 2024

			Array begins:
+-----+-----------------------------------------------+
| n\k |   2    3     4    5      6     7       8  ... |
+-----+-----------------------------------------------+
|  0  |   1    2     3    4      5     6       7  ... |
|  1  |   1    4     9   16     25    36      49  ... |
|  2  |   1    2    27   32    125    27     343  ... |
|  3  |   1    8    27  128    625   216    2401  ... |
|  4  |   1    1    81  128   3125    81   16807  ... |
|  5  |   1    4   243  512   3125   972  117649  ... |
|  6  |   1    4   243 1024  15625   486  823543  ... |
|  7  |   1   16   729 4096  78125 11664  823543  ... |
|  8  |   1    1  2187 2048 390625  2187 5764801  ... |
| ... | ...  ...   ...  ...    ...   ...     ...  ... |
+-----+-----------------------------------------------+

Cf. A374895 (numerators).

Cf. A008292, A121985, A277542.

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

T(n,k) = denominator(polylog(-n, 1/k)).
T(n,k) = denominator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
T(0,k) = k-1.
T(1,k) = (k-1)^2.
T(2,k) = A277542(k-1).
T(n,2) = 1.
T(n,n) = A121985(n).

cp's OEIS Frontend

A121376 Numerator of PolyLog(-n, 1/n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula