A009362 Expansion of log(1 + sinh(x)/exp(x)).

0, 1, -3, 12, -66, 480, -4368, 47712, -608016, 8855040, -145083648, 2641216512, -52891055616, 1155444326400, -27344999497728, 696933753434112, -19031293222127616, 554336947975618560, -17155693983744196608

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Crossrefs

Cf. A008292 (Eulerian numbers).

Programs

Mathematica

Log[ 1+Sinh[ x ]/Exp[ x ] ]
CoefficientList[Series[Log[1 + Sinh[x]/E^x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)

PARI

a(n)=-(-1)^n*sum(k=0,n-1,3^k*sum(j=0,k,(-1)^j*(k-j)^(n-1)*binomial(n,j))) \\ Paul D. Hanna, Mar 29 2006

Formula

a(n) = -Sum_{k>0} (-2*k)^n/3^k/k = -(-2)^n*polylog(-n+1, 1/3), n>0. - Vladeta Jovovic, Sep 30 2003

a(n) = -(-1)^n*Sum_{k=0..n-1} 3^k*Sum_{j=0..k} (-1)^j*(k-j)^n*C(n,j) for n>0. a(n) = -(-1)^n*Sum_{k=0..n-1} 3^k*A008292(n-1,k) for n>0, where A008292 are the Eulerian numbers. - Paul D. Hanna, Mar 29 2006

a(n) ~ (n-1)! * (-1)^(n+1) * (2/log(3))^n. - Vaclav Kotesovec, Jan 23 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997