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
Examples
PolyLog(-n, 1/n) begins -1/12, 6, 33/8, 380/81, 3535/512, 189714/15625, ...
Links
- Thomas Scheuerle, Table of n, a(n) for n = 1..150
- Eric Weisstein's World of Mathematics, Polylogarithm.
Programs
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Mathematica
Numerator[Table[PolyLog[ -n,1/n],{n,1,40}]] -
PARI
a(n)=if(n==1,-1,numerator(polylog(-n,1/n))) \\ Charles R Greathouse IV, Jul 14 2014
Formula
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