A234290 Duplicate of A095839.

1, 1, 5, 51, 807, 17445, 479565, 16019955, 630301455, 28552506885, 1463744449125, 83780913568275, 5296205435649975, 366478026602012325, 27552067849812030525, 2236327624673777509875, 194908916445067162713375, 18154937081288124469477125, 1799824448875247911270279125

Offset: 0

Author

Paul D. Hanna, Dec 22 2013

Keywords

• dead

Comments

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.

Is this sequence the same as A095839?

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 51*x^3/3! + 807*x^4/4! + 17445*x^5/5! +...
where A(x)^3 = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4275*x^4/4! + 97155*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - 3*x^3/3! - 27*x^4/4! - 405*x^5/5! - 8505*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 3*x^3/3 + 12*x^4/4 + 55*x^5/5 + 273*x^6/6 + 1428*x^7/7 +...+ A001764(n-1)*x^n/n +...
where A001764(n) = binomial(3*n,n)/(2*n+1).
		

Crossrefs

Cf. A058562, A234289, A234291, A001764, A000108.

Programs

Mathematica

CoefficientList[Series[(2-Sqrt[1-6*x])/(1+2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

PARI

{a(n)=local(A=1+x);for(i=1,n,A=1+A^3*intformal(1/(A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

PARI

/* From formula using g.f. of A001764 G(x) = 1 + x*G(x)^3: */
{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/deriv(serreverse(intformal(G))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* Explicit formula: 1/(1 - x*C(3*x/2)), C(x) = 1 + x*C(x)^2 */
{a(n)=local(A=(2-sqrt(1-6*x+x^2*O(x^n)))/(1+2*x)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* From formula: 1 + Series_Reversion((x - x^2/2)/(1+x)^2): */
{a(n)=local(A=1,X=x+x^2*O(x^n));A=1+serreverse((X-X^2/2)/(1+X)^2); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x) dx ) ) where G(x) = 1 + x*G(x)^3 = Sum_{n>=0} A001764(n)*x^n is the g.f. of A001764.

E.g.f.: (2 - sqrt(1-6*x)) / (1+2*x) = 1/(1 - x*C(3*x/2)), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

E.g.f.: 1 + Series_Reversion( (x - x^2/2) / (1+x)^2 ).

a(n) ~ 2^(n-5/2) * 3^(n+1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013

D-finite with recurrence a(n) +(-4*n+9)*a(n-1) -6*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024