OFFSET
0,3
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
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
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 *)
PROG
(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;
(PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(-x) ))) \\ G. C. Greubel, Jan 21 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Extended with signs by Olivier Gérard, Mar 15 1997
STATUS
approved