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

0,4

Links

Table of n, a(n) for n=0..21.

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

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A069814.

Sequence in context: A151490 A282475 A185652 * A097574 A113023 A174082

Adjacent sequences: A009302 A009303 A009304 * A009306 A009307 A009308

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 21 2011

Status

approved