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A058562 Another 3-way generalization of series-parallel networks with n labeled edges.

%I A058562 #50 Jan 11 2025 13:12:39
%S A058562 0,1,3,21,243,3933,81819,2080053,62490339,2166106509,85092601707,
%T A058562 3735939709989,181287330220467,9634718677393917,556569415611455931,
%U A058562 34723276781195740437,2326773811332029313411,166666995789875216053101,12708546598923724476443403
%N A058562 Another 3-way generalization of series-parallel networks with n labeled edges.
%H A058562 Alois P. Heinz, <a href="/A058562/b058562.txt">Table of n, a(n) for n = 0..360</a>
%H A058562 O. Bodini, A. Genitrini, F. Peschanski and N. Rolin, <a href="http://dx.doi.org/10.1007/978-3-319-14974-5_21">Associativity for binary parallel processes</a>, CALDAM 2015; [<a href="http://caldam.cse.iitk.ac.in/download.php?file=presentations/day2/session7/1-bogerope.pdf">Slides</a>]
%F A058562 E.g.f.: -3/2*LambertW(-2/3*exp(-2/3+1/3*x))-1. - _Vladeta Jovovic_, Jun 25 2007
%F A058562 E.g.f.: A(x) = Series_Reversion[ 3*log(1+x) - 2*x ]. [_Paul D. Hanna_, Aug 03 2008]
%F A058562 Let f(x) = (1+x)/(1-2*x). Let D be the operator g(x) -> d/dx(f(x)*g(x)). Then for n>=1, a(n) = D^(n-1)(1) evaluated at x = 0. - _Peter Bala_, Sep 05 2011
%F A058562 log(1 + A(x)) = x + 2*x^2/2! + 14*x^3/3! + 162*x^4/4! + ... is the e.g.f. for A201465. - _Peter Bala_, Jul 12 2012
%F A058562 a(n) = sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (3^(j-l)*(2)^l*(-1)^(l+j)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!)))).  [_Vladimir Kruchinin_, Sep 26 2012]
%F A058562 G.f.: x/Q(0), where Q(k)= 1 - (k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013
%F A058562 a(n) ~ sqrt(3) * n^(n-1) / (2*exp(n) * (log(27/8)-1)^(n-1/2)). - _Vaclav Kotesovec_, Oct 05 2013
%F A058562 a(n) = a(n-1) + Sum_{j=1..n-1} binomial(n,j)*a(j)*a(n-j) for n>1. - _Peter Luschny_, May 24 2017
%p A058562 spec := [ N, {N=Union(Z,S,P,Q), S=Set(Union(Z,P,Q),card>=2), P=Set(Union(Z,S,Q),card>=2), Q=Set(Union(Z,S,P),card>=2)}, labeled ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N=A058562, S=A058575
%p A058562 # Alternatively:
%p A058562 A058562_list := proc(len) local A, n; A[0] := 0; A[1] := 1; for n from 2 to len do
%p A058562 A[n] := A[n-1] + add(binomial(n,j)*A[j]*A[n-j], j=1..n-1) od:
%p A058562 convert(A,list) end: A058562_list(18); # _Peter Luschny_, May 24 2017
%t A058562 a[n_] := Sum[(n+k-1)!*Sum[1/(k-j)!*Sum[(3^(j-l)*(2)^l*(-1)^(l+j)* StirlingS1[n-l+j-1, j-l])/(l!*(n-l+j-1)!), {l, 0, j}], {j, 0, k}], {k, 0, n-1}]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)
%o A058562 (PARI) {a(n)=if(n<1,0,n!*polcoeff(serreverse(3*log(1+x+x*O(x^n))-2*x),n))} \\ _Paul D. Hanna_, Aug 03 2008
%o A058562 (Maxima) a(n):=sum((n+k-1)!*sum(1/(k-j)!*sum((3^(j-l)*(2)^l*(-1)^(l+j)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!),l,0,j),j,0,k),k,0,n-1); /* _Vladimir Kruchinin_, Sep 26 2012 */
%Y A058562 Cf. A058540, A058371, A058575, A201465.
%K A058562 nonn
%O A058562 0,3
%A A058562 _N. J. A. Sloane_, Dec 26 2000