A234291 - OEIS

%I #19 Dec 27 2013 14:05:09

%S 1,1,4,34,460,8608,206152,6020992,207574240,8251015264,371527296256,

%T 18691127602816,1039066330203520,63253339835514112,

%U 4184830238170091008,298985971407749744128,22941517126450315985920,1881603821848104123344896,164271703613261014954276864

%N E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x)^2 dx.

%C Compare to: G(x) = 1 + G(x)^3 * Integral 1/G(x)^3 dx, where G(x)-1 is the e.g.f. of A058562, the number of 3-way series-parallel networks with n labeled edges.

%F E.g.f.: 1 / sqrt( d/dx Series_Reversion( Integral G(x)^2 dx ) ) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764, and G(x)^2 is the g.f. of A006013.

%F E.g.f.: 1 + Series_Reversion( 3*x/(1+x) - 2*log(1+x) ).

%F E.g.f.: -3/(2*LambertW(-1,-3/2*exp((x-3)/2))). - _Vaclav Kotesovec_, Dec 27 2013

%F a(n) ~ 3*sqrt(2) * n^(n-1) / (4*exp(n) * (1+2*log(2)-2*log(3))^(n-1/2)). - _Vaclav Kotesovec_, Dec 27 2013

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 460*x^4/4! + 8608*x^5/5! +...

%e where A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2628*x^4/4! + 51264*x^5/5! +...

%e Integral 1/A(x)^2 dx = x - 2*x^2/2! - 2*x^3/3! - 20*x^4/4! - 272*x^5/5! - 5096*x^6/6! +...

%e Further,

%e Series_Reversion(Integral 1/A(x)^2 dx) = x + 2*x^2/2 + 7*x^3/3 + 30*x^4/4 + 143*x^5/5 + 728*x^6/6 + 3876*x^7/7 +...+ A006013(n-1)*x^n/n +...

%e where A006013(n) = binomial(3*n+1,n)/(n+1).

%p seq(n! * coeff(series(-3/(2*LambertW(-1,-3/2*exp((x-3)/2))), x, n+1), x, n), n = 0..10) # _Vaclav Kotesovec_, Dec 27 2013

%t CoefficientList[1 + InverseSeries[Series[3*x/(1+x) - 2*Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 27 2013 *)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^3*intformal(1/(A^2+x*O(x^n)))); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* Explicit formula using g.f. of A001764, G(x) = 1 + x*G(x)^3: */

%o {a(n)=local(G=sum(m=0, n, binomial(3*m, m)/(2*m+1)*x^m)+x*O(x^n), A=1); A=1/sqrt(deriv(serreverse(intformal(G^2)))); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* Explicit formula: 1 + Series_Reversion(3*x/(1+x) - 2*log(1+x)): */

%o {a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(3*X/(1+X)-2*log(1+X)); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A001764, A006013, A058562, A234290, A234289.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 22 2013