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A058562
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Another 3-way generalization of series-parallel networks with n labeled edges.
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7
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0, 1, 3, 21, 243, 3933, 81819, 2080053, 62490339, 2166106509, 85092601707, 3735939709989, 181287330220467, 9634718677393917, 556569415611455931, 34723276781195740437, 2326773811332029313411, 166666995789875216053101, 12708546598923724476443403
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: -3/2*LambertW(-2/3*exp(-2/3+1/3*x))-1. - Vladeta Jovovic, Jun 25 2007
E.g.f.: A(x) = Series_Reversion[ 3*log(1+x) - 2*x ]. [Paul D. Hanna, Aug 03 2008]
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
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
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]
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
a(n) ~ sqrt(3) * n^(n-1) / (2*exp(n) * (log(27/8)-1)^(n-1/2)). - Vaclav Kotesovec, Oct 05 2013
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
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MAPLE
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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
# Alternatively:
A058562_list := proc(len) local A, n; A[0] := 0; A[1] := 1; for n from 2 to len do
A[n] := A[n-1] + add(binomial(n, j)*A[j]*A[n-j], j=1..n-1) od:
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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