OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..750
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( x*(A(x) + 2*A(x)^2) + Integral(A(x) + 2*A(x)^2 dx) ).
(2) A(x) = (1/x)*Series_Reversion( x*(1-4*x-2*x^2)/(1+x)^2 ).
(3) A(x) = 1 + x*A(x)*(1 + 2*A(x))*(2 + x*A(x)).
(4) A(x) = 1 + Sum_{n>=2} (-1)^n * 3*n * x^(n-1) * A(x)^n.
Recurrence: 8*n*(n+1)*(19*n-25)*a(n) = 4*n*(798*n^2 - 1449*n + 529)*a(n-1) - (513*n^3 - 1701*n^2 + 1630*n - 432)*a(n-2) + 2*(n-3)*(2*n-3)*(19*n-6)*a(n-3). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ 2^(2*n-5/2) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 0.1919459762582734141... is the root of the equation -128 + 672*r - 27*r^2 + r^3 = 0 and s = 0.0229965315195763941... is the root of the equation -19 + 798*s + 1215*s^2 + 512*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013
EXAMPLE
G.f.: A(x) = 1 + 6*x + 63*x^2 + 822*x^3 + 12000*x^4 + 187632*x^5 + ...
Related expansions:
(1 + x*A(x))^2 = 1 + 2*x + 13*x^2 + 138*x^3 + 1806*x^4 + 26400*x^5 + ...
(1 + 2*A(x))/3 = 1 + 4*x + 42*x^2 + 548*x^3 + 8000*x^4 + 125088*x^5 + ...
A(x) + 2*A(x)^2 = 3 + 30*x + 387*x^2 + 5622*x^3 + 87666*x^4 + 1433304*x^5 + ...
log(A(x)) = 6*x + 90*x^2/2 + 1548*x^3/3 + 28110*x^4/4 + 525996*x^5/5 + ...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x*(1-4*x-2*x^2)/(1+x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *)
PROG
(PARI) {a(n)=polcoeff((serreverse(x*(1-4*x-2*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+2*A^2)+intformal(A+2*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+2*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