A201465 -id:A201465 - cp's OEIS frontend

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

0, 1, 3, 21, 243, 3933, 81819, 2080053, 62490339, 2166106509, 85092601707, 3735939709989, 181287330220467, 9634718677393917, 556569415611455931, 34723276781195740437, 2326773811332029313411, 166666995789875216053101, 12708546598923724476443403

Offset: 0

N. J. A. Sloane, Dec 26 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..360
O. Bodini, A. Genitrini, F. Peschanski and N. Rolin, Associativity for binary parallel processes, CALDAM 2015; [Slides]

Cf. A058540, A058371, A058575, A201465.

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:
convert(A,list) end: A058562_list(18); # Peter Luschny, May 24 2017

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 *)

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 */

{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

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

A201466 E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.

Original entry on oeis.org

1, 3, 30, 498, 11568, 345432, 12606240, 543678672, 27054328512, 1525746223488, 96167433279360, 6699404849841408, 511152613463843328, 42391161255859802112, 3796840836492517125120, 365260399012767192102912, 37561729737177160757133312, 4111876748834828077514170368

Offset: 1

Paul D. Hanna, Dec 01 2011

nonn

			E.g.f.: A(x) = x + 3*x^2/2! + 30*x^3/3! + 498*x^4/4! + 11568*x^5/5! + 345432*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! + 460576*x^6/6! +...
where x = 3 + 4*A(x) - 3*exp(A(x)).
...
O.g.f.: G(x) = 1 + 3*x + 30*x^2 + 498*x^3 + 11568*x^4 + 345432*x^5 +...
where
G(x) = 1/4 + 3/(4*(4-x)) + 3^2/(4*(4-x)*(4-2*x)) + 3^3/(4*(4-x)*(4-2*x)*(4-3*x)) + 3^4/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)) + 3^5/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)*(4-5*x)) +...

Cf. variants: A000311, A201465.

Rest[CoefficientList[InverseSeries[Series[3 - 3*E^x + 4*x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=n!*polcoeff(serreverse(3+4*x - 3*exp(x+x^2*O(x^n))),n)}

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*3^n/prod(k=0, n, 4 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. satisfies: x = A( 3 + 4*x - 3*exp(x) ).
E.g.f.: (x-3)/4 - LambertW(-3*exp((x-3)/4)/4). - Vaclav Kotesovec, Jan 10 2014
a(n) ~ n^(n-1) / (2 * (4*log(4/3)-1)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Jan 10 2014
O.g.f.: Sum_{n>=0} 3^n / Product_{k=0..n} (4 - k*x). - Paul D. Hanna, Oct 27 2014

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

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A201466 E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.

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