A250918 G.f. A(x) satisfies: 2*A(x) = 1 + A(x)^3/(1 + x*A(x)^3).

1, 1, 2, 6, 19, 67, 244, 928, 3608, 14348, 57936, 237228, 982006, 4104266, 17292220, 73373704, 313255939, 1344703363, 5800347378, 25128400374, 109287352255, 476990070863, 2088544907796, 9171761291584, 40385942346364, 178272020615752, 788731516970684, 3496994698151688, 15535135825359760, 69140216020170812

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f. A(x) satisfies:

(1) 1 - 2*A(x) + (1+x)*A(x)^3 - 2*x*A(x)^4 = 0.

(2) (1 - 2*A(x) + A(x)^3) / (2*A(x)^4 - A(x)^3) = x.

(3) 1/(1 + x*A(x)^3) + 2*x*A(x) = 1 + x.

(4) A(x) = 1 + Series_Reversion( x*(1 + 3*x + x^2) / ((1+x)^3 * (1+2*x)) ).

a(n) ~ sqrt(8 - 5*sqrt(2/3) + 3*sqrt(9-3*sqrt(6))) / (4 * n^(3/2) * sqrt(Pi) * (2*sqrt(6) - 5 + 4/sqrt(81 + 33*sqrt(6)))^n). - Vaclav Kotesovec, Jan 17 2015

Example

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 67*x^5 + 244*x^6 + 928*x^7 +...
where
2*A(x) = 1 + A(x)^3 - x*A(x)^6 + x^2*A(x)^9 - x^3*A(x)^12 + x^4*A(x)^15 - x^5*A(x)^18 + x^6*A(x)^21 - x^7*A(x)^24 +-...
Also
2*A(x) = 1 + (1+x)*A(x)^3 - 2*x*A(x)^4.

Mathematica

CoefficientList[1 + InverseSeries[Series[x*(1 + 3*x + x^2) / ((1+x)^3 * (1+2*x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Jan 17 2015 *)

Prog

(PARI) /* Generates vector A of initial terms: */
{A=[1, 1]; for(i=1, 40, A=concat(A, 0); A[#A]=Vec(sum(n=1, #A, (-x)^n*Ser(A)^(3*n)))[#A]); A}
(PARI) /* G.f.: 1 + Series_Reversion(x*(1+3*x+x^2)/((1+x)^3*(1+2*x))) */
{a(n)=local(A= 1 + serreverse(x*(1 + 3*x + x^2)/((1+x)^3*(1+2*x)) +x^2*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A234010 A360212 A121655 * A150095 A150096 A150097

Adjacent sequences: A250915 A250916 A250917 * A250919 A250920 A250921

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2015

Status

approved