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A234290 E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x) dx. 2
1, 1, 5, 51, 807, 17445, 479565, 16019955, 630301455, 28552506885, 1463744449125, 83780913568275, 5296205435649975, 366478026602012325, 27552067849812030525, 2236327624673777509875, 194908916445067162713375, 18154937081288124469477125, 1799824448875247911270279125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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?

LINKS

Table of n, a(n) for n=0..18.

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

EXAMPLE

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).

MATHEMATICA

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

PROG

(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), ", "))

CROSSREFS

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

Sequence in context: A318192 A299435 A095839 * A107669 A218675 A182316

Adjacent sequences:  A234287 A234288 A234289 * A234291 A234292 A234293

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 22 2013

STATUS

approved

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Last modified October 23 03:21 EDT 2019. Contains 328335 sequences. (Running on oeis4.)