OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..680
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( x*(A(x) + 3*A(x)^2) + Integral(A(x) + 3*A(x)^2 dx) ).
(2) A(x) = (1/x)*Series_Reversion( x*(1-6*x-3*x^2)/(1+x)^2 ).
(3) A(x) = 1 + x*A(x)*(1 + 3*A(x))*(2 + x*A(x)).
(4) A(x) = 1 + Sum_{n>=2} (-1)^n * 4*n * x^(n-1) * A(x)^n.
Recurrence: 6*n*(n+1)*(85*n-111)*a(n) = 3*n*(4930*n^2 - 8903*n + 3231)*a(n-1) - 2*(850*n^3 - 2810*n^2 + 2679*n - 702)*a(n-2) + (n-3)*(2*n-3)*(85*n-26)*a(n-3). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ 18^(n-1) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 0.6231608238452908396... is the root of the equation -17496 + 28188*r - 180*r^2 + r^3 = 0 and s = 0.00248396272197254381... is the root of the equation -85 + 33660*s + 224208*s^2 + 419904*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013
EXAMPLE
G.f.: A(x) = 1 + 8*x + 116*x^2 + 2096*x^3 + 42396*x^4 + 918648*x^5 + ...
Related expansions:
(1 + x*A(x))^2 = 1 + 2*x + 17*x^2 + 248*x^3 + 4488*x^4 + 90840*x^5 + ...
(1 + 3*A(x))/4 = 1 + 6*x + 87*x^2 + 1572*x^3 + 31797*x^4 + 688986*x^5 + ...
A(x) + 3*A(x)^2 = 4 + 56*x + 1004*x^2 + 20240*x^3 + 437748*x^4 + ...
log(A(x)) = 8*x + 168*x^2/2 + 4016*x^3/3 + 101200*x^4/4 + 2626488*x^5/5 + ...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x*(1-6*x-3*x^2)/(1+x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *)
PROG
(PARI) {a(n)=polcoeff((serreverse(x*(1-6*x-3*x^2)/(1+x)^2 +x^2*O(x^n))/x), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(A+3*A^2)+intformal(A+3*A^2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=1+x*A*(1+3*A)*(2+x*A) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 10 2013
STATUS
approved