A009578 E.g.f. sinh(log(1+x))/exp(x). Unsigned sequence gives degrees of (finite by nilpotent) representations of Braid groups.

0, 1, -3, 9, -34, 165, -981, 6853, -54804, 493209, -4932055, 54252561, -651030678, 8463398749, -118487582409, 1777313736045, -28437019776616, 483429336202353, -8701728051642219, 165332832981202009, -3306656659624040010

Offset: 0

Author

R. H. Hardin, Stephen P. Humphries

Keywords

• sign
• easy

Comments

Unsigned sequence gives number of undirected paths through K_n, and including paths with zero edges. - Thomas Anton, Jan 14 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A009132.

Programs

Magma

f := function(n) nn := n; for i := 2 to n do nn := nn+Factorial(n)/(Factorial(n-i)*2); end for; return nn; end function;

Maple

g:=(1/2)*x*(2+x)*exp(-x)/(1+x): gser:=series(g,x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Nov 16 2007

Mathematica

With[{nmax=30}, CoefficientList[Series[Sinh[Log[1+x]]*Exp[-x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jan 21 2018 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(-x) ))) \\ G. C. Greubel, Jan 21 2018

Formula

a(n) = n*(1 + A000522(n-1))/2 for n > 0.

For the unsigned sequence, a(n) = n*(2*a(n-1) + 3 - n)/2, a(0)=0. - Emeric Deutsch, Oct 24 2001

a(n) = (-1)^(n-1)/2*floor(n!*exp(1) + n-1), n > 0. - Vladeta Jovovic, Aug 17 2002

The defining generating function simplifies to x*(2+x)*exp(-x)/(2*(1+x)). - Emeric Deutsch, Nov 16 2007

E.g.f.: 1/2*U(0) where U(k)= 1 + x/(1 + 1/(x*(x+1) - 1 - x^2*(x+1)^2/(x*(x+1) + (k+1)/U(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Nov 16 2012

D-finite with recurrence a(n) +(n+3)*a(n-1) +3*n*a(n-2) +(3*n-5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Aug 15 2025

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997