A104209 - OEIS

%I #24 Jan 13 2022 05:37:20

%S 1,3,39,819,23949,898947,41212155,2232057171,139455901101,

%T 9873341493231,781184921112075,68309191570851759,6541702440222052137,

%U 680922615974259589527,76544749927261960908807,9241807764375868372683255,1192762017796744530286451865

%N Number of labeled directed multigraphs with n arrows and no vertex of degree 0.

%C These are the dimensions of the homogeneous components of a commutative graded Hopf algebra generalizing quasi-symmetric functions.

%H J.-C. Novelli, J.-Y. Thibon and N. M. Thiéry, <a href="http://dx.doi.org/10.1016/j.crma.2004.09.012">Algèbres de Hopf de graphes</a> [Hopf algebras of graphs], C.R. Acad. Sci. Paris (Comptes Rendus Mathématique), 339 (2004), 607-610.

%F a(n) = Sum_{m >=0} binomial(m^2+n-1, n)/2^(m+1).

%F G.f.: Sum_{m >= 0} (1-x)^(-m^2)/2^(m+1). Row sums of A120945. - _Vladeta Jovovic_, Sep 25 2006

%F a(n) ~ c * 2^(2*n) * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.432167265869761794333243584356866417673557873163120324347... = 2^(log(2)/8 - 1) / (sqrt(Pi) * log(2)). - _Vaclav Kotesovec_, May 03 2015, updated Mar 21 2018

%e a(1)=3, the three graphs being (1 -> 2), (2 -> 1) and (1 -> 1).

%p d:=proc(n) local m;sum(binomial(m^2+n-1,n)/2^(m+1),m=0..infinity);end;

%t f[n_] := Sum[ Binomial[m^2 + n - 1, n]/2^(m + 1), {m, 0, Infinity}]; Table[ f[n], {n, 0, 15}] (* _Robert G. Wilson v_, Mar 16 2005 *)

%t Table[Sum[Sum[(-1)^(k-j)*Binomial[k,j]*Binomial[j^2+n-1,n],{j,0,k}],{k,0,2*n}],{n,0,20}] (* _Vaclav Kotesovec_, May 03 2015, much faster *)

%Y Cf. A052171 (counts same objects up to labeling).

%Y Cf. A020561, A120945.

%K nonn

%O 0,2

%A Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Mar 13 2005

%E Corrected and extended by _Robert G. Wilson v_, Mar 16 2005

%E Offset corrected by _Vaclav Kotesovec_, May 03 2015