A245043 G.f. satisfies: A(x) = (12 + A(x)^4) / (13 - 27*x).

1, 3, 15, 117, 1158, 12930, 154986, 1947582, 25317009, 337610451, 4592807895, 63488144109, 889226772132, 12592147132572, 179982549300948, 2593187073716460, 37622924436008574, 549178914689660106, 8059539548880228138, 118846096104074358942, 1760035125442960123992

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f. satisfies:

(1) A(x) = 1 + Series_Reversion( (1+13*x - (1+x)^4)/(27*(1+x)) ).

(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x*A(x))^(3*n+1) / 13^(4*n+1).

(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (12+27*x + G(x)^4)/13 is the g.f. of A120595.

a(n) ~ 3^(3*n - 7/2) / (2^(7/4) * sqrt(Pi) * n^(3/2) * (13 - 8*sqrt(2))^(n - 3/2)). - Vaclav Kotesovec, Nov 27 2017

Example

G.f.: A(x) =  1 + 3*x + 15*x^2 + 117*x^3 + 1158*x^4 + 12930*x^5 +...
Compare A(x)^4 to (13-27*x)*A(x):
A(x)^4 = 1 + 12*x + 114*x^2 + 1116*x^3 + 11895*x^4 + 136824*x^5 +...
(13-27*x)*A(x) = 13 + 12*x + 114*x^2 + 1116*x^3 + 11895*x^4 + 136824*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/(27*(1+x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)

Prog

(PARI) {a(n)=polcoeff(1 + serreverse( (1+13*x - (1+x)^4)/(27*(1+x +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^4 - (13-27*x)*Ax )/9 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A120595, A245009, A244627, A244594, A244856.

Sequence in context: A341647 A074596 A087806 * A225109 A080290 A371984

Adjacent sequences: A245040 A245041 A245042 * A245044 A245045 A245046

Keyword

nonn

Author

Paul D. Hanna, Jul 10 2014

Status

approved