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A231615 G.f. satisfies: A(x) = (1 - 2*x*A(x))^2 * (3*A(x) - 2). 9
1, 2, 14, 116, 1068, 10512, 108288, 1152944, 12586256, 140119328, 1584718432, 18156868096, 210302739712, 2458400698368, 28966931629056, 343671253924608, 4102106153148672, 49225440239052288, 593522724752742912, 7186802279959262208, 87357857306307234816, 1065563386236346036224 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..800

FORMULA

G.f. A(x) satisfies:

(1) A(x) = exp( x*(3*A(x)^2 - 2*A(x)) + Integral(3*A(x)^2 - 2*A(x) dx) ).

(2) A(x) = (1/x)*Series_Reversion( x*(1-6*x+6*x^2)/(1-2*x)^2 ).

(3) A(x) = 1 + 2*x*A(x)*(3*A(x) - 2)*(1 - x*A(x)).

(4) A(x) = 1 + Sum_{n>=2} 2^(n-2) * n * x^(n-1) * A(x)^n.

Recurrence: 3*n*(n+1)*(41*n-51)*a(n) = 3*n*(574*n^2 - 1001*n + 369)*a(n-1) - (1517*n^3 - 4921*n^2 + 4530*n - 1080)*a(n-2) + 8*(n-3)*(2*n-3)*(41*n-10)*a(n-3). - Vaclav Kotesovec, Dec 29 2013

a(n) ~ 2^(5*n-3/2) * 3^(2*n-5/2) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 22.00345492406548979... is the root of the equation -4478976 + 217728*r - 666*r^2 + r^3 = 0 and s = 0.00113473758620755613... is the root of the equation -41 + 30996*s + 4320054*s^2 + 181398528*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013

EXAMPLE

G.f.: A(x) = 1 + 2*x + 14*x^2 + 116*x^3 + 1068*x^4 + 10512*x^5 +...

Related expansions.

(1 - 2*x*A(x))^2 = 1 - 4*x - 4*x^2 - 40*x^3 - 336*x^4 - 3120*x^5 -...

3*A(x) - 2 = 1 + 6*x + 42*x^2 + 348*x^3 + 3204*x^4 + 31536*x^5 +...

3*A(x)^2 - 2*A(x) = 1 + 8*x + 68*x^2 + 632*x^3 + 6252*x^4 + 64608*x^5 +...

log(A(x)) = 2*x + 24*x^2/2 + 272*x^3/3 + 3160*x^4/4 + 37512*x^5/5 +...

MATHEMATICA

CoefficientList[1/x*InverseSeries[Series[x*(1-6*x+6*x^2)/(1-2*x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *)

PROG

(PARI) {a(n)=polcoeff((serreverse(x*(1-6*x+6*x^2)/(1-2*x)^2 +x^2*O(x^n))/x), n)}

for(n=0, 30, print1(a(n), ", "))

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

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n)=local(A=1); for(i=1, n, A=1+2*x*A*(3*A-2)*(1-x*A) +x*O(x^n)); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A228966, A231552, A231553, A231554, A231556, A231616, A231618.

Sequence in context: A192406 A332664 A092639 * A277467 A208393 A192457

Adjacent sequences:  A231612 A231613 A231614 * A231616 A231617 A231618

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 11 2013

STATUS

approved

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Last modified October 4 08:40 EDT 2022. Contains 357239 sequences. (Running on oeis4.)