A009305 Expansion of e.g.f. log(1 + x*cosh(x)).

0, 1, -1, 5, -18, 89, -600, 4717, -42896, 449073, -5287680, 69090581, -993391872, 15583801609, -264816161792, 4846181282685, -95022445824000, 1987373846425697, -44163232640630784, 1039121484066627877, -25807915421845422080, 674707915373741222841

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Crossrefs

Cf. A069814.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Log[1+Cosh[x]*x],{x,0,nn}],x]Range[ 0,nn]!] (* Harvey P. Dale, Mar 09 2013 *)

Maxima

a(n):=n!*sum(sum((k-2*i)^(n-k)*binomial(k,i),i,0,k)/(2^k*(n-k)!)*(-1)^(k-1)/k,k,1,n-1)+(-1)^(n-1)*(n-1)!; /* Vladimir Kruchinin, Apr 21 2011 */

PARI

x='x+O('x^66); /* that many terms */
egf=log(1+x*cosh(x)); /* = x - 1/2*x^2 + 5/6*x^3 - 3/4*x^4 + 89/120*x^5 +-... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 21 2011 */

Formula

a(n) = n!*Sum_{k=1..n-1} ((Sum_{i=0..k} (k-2*i)^(n-k)*binomial(k,i)) /(2^k*(n-k)!)*(-1)^(k-1)/k) + (-1)^(n-1)*(n-1)!. - Vladimir Kruchinin, Apr 21 2011

a(n) ~ (n-1)! * (-1)^(n+1) / r^n, where r = 0.765009954550732122655321742482815219200352137475... (see A069814) is the root of the equation r*cosh(r) = 1 . - Vaclav Kotesovec, Jan 24 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 21 2011