A052832 Expansion of e.g.f. log((-1+x)/(-1+x+x^2)).

0, 0, 2, 6, 36, 240, 2040, 20160, 231840, 3024000, 44271360, 718502400, 12813292800, 249080832000, 5243151513600, 118824010905600, 2884729655808000, 74694359900160000, 2054806272110592000, 59849389401145344000, 1840003788783992832000, 59545276650123264000000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Old name was: A simple grammar.

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 797

Programs

Maple

spec := [S,{B=Prod(Z,C),C=Sequence(Z,1 <= card),S= Cycle(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052832 := proc(n)
    log((-1+x)/(-1+x+x^2));
    coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052832(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Formula

Recurrence: {a(1)=0, a(3)=6, a(2)=2, (n^3+3*n^2+2*n)*a(n)+(-4-2*n)*a(n+2)+a(n+3)}.

(RootOf(_Z^2-_Z-1)^n*RootOf(_Z^2-_Z-1)+(1-RootOf(_Z^2-_Z-1))^(n+1)-1)*GAMMA(n+1)/RootOf(_Z^2-_Z-1)/(-1+RootOf(_Z^2-_Z-1)).

D-finite with recurrence a(n) +2*(-n+1)*a(n-1) +(n-1)*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 13 2025

Extensions

More terms from Alois P. Heinz, Mar 16 2016