A244594 G.f. satisfies: A(x) = (3 + A(x)^3) / (4 - x).

1, 1, 4, 29, 263, 2672, 29088, 331749, 3912660, 47329811, 583983656, 7321173872, 92990672635, 1194113490556, 15476763809428, 202197552311829, 2659975668005367, 35205831900984144, 468468683002725372, 6263539340729569047, 84103985900174324256, 1133671250214654009000

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

G.f. satisfies:

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

(2) A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (3 + x*A(x))^(2*n+1) / 4^(3*n+1).

(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (3+x + G(x)^3)/4 is the g.f. of A120590.

Recurrence: 13*(n-1)*n*a(n) = 96*(n-1)*(2*n-3)*a(n-1) - 8*(6*n^2 - 24*n + 23)*a(n-2) + 2*(n-2)*(2*n-7)*a(n-3). - Vaclav Kotesovec, Jul 03 2014

a(n) ~ sqrt(48-18^(4/3)) * ((24*18^(1/3)+9*18^(2/3)+64)/13)^n / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2014

Example

G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 263*x^4 + 2672*x^5 + 29088*x^6 +...
Compare
(4 - x)*A(x) = 4 + 3*x + 15*x^2 + 112*x^3 + 1023*x^4 + 10425*x^5 +...
to:
A(x)^3 = 1 + 3*x + 15*x^2 + 112*x^3 + 1023*x^4 + 10425*x^5 + 113680*x^6 +...

Prog

(PARI) {a(n)=polcoeff(1+serreverse((1+4*x - (1+x)^3)/(1 + x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1, 1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( Ax^3 - (4-x)*Ax )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A120590, A001764.

Sequence in context: A203970 A250885 A369215 * A168238 A370955 A294160

Adjacent sequences: A244591 A244592 A244593 * A244595 A244596 A244597

Keyword

nonn

Author

Paul D. Hanna, Jul 01 2014

Status

approved