A245009 G.f. satisfies: A(x) = (7 + A(x)^4) / (8 - 8*x).

1, 2, 10, 88, 978, 12200, 163156, 2286448, 33138874, 492657384, 7470940300, 115115319376, 1797128902132, 28364816229008, 451870965523368, 7256283996155360, 117333885356923274, 1908844190372949224, 31221135850863938268, 513100005743085437328, 8468653781083527106012, 140314257925457275837488

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f. satisfies:

(1) A(x) = 1 + Series_Reversion( (1+8*x - (1+x)^4)/(8*(1+x)) ).

(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7 + 8*x*A(x))^(3*n+1) / 8^(4*n+1).

(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (7+8*x + G(x)^4)/8 is the g.f. of A120594.

a(n) ~ 3^(3*(n-1)/4) * 7^((n-1)/4) / (sqrt(Pi) * n^(3/2) * (3^(3/4)*7^(1/4) - 7/2)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017

Example

G.f.: A(x) = 1 + 2*x + 10*x^2 + 88*x^3 + 978*x^4 + 12200*x^5 +...
Compare A(x)^4 to 8*(1-x)*A(x):
A(x)^4 = 1 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...
8*(1-x)*A(x) = 8 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/(8*(1+x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)

Prog

(PARI) {a(n)=polcoeff(1 + serreverse((1+8*x - (1+x)^4)/(8*(1+x +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1], Ax=1+2*x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^4 - 8*(1-x)*Ax )/4 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A120594, A244627, A244594, A244856, A245043.

Sequence in context: A295836 A245496 A185388 * A354240 A306404 A111811

Adjacent sequences: A245006 A245007 A245008 * A245010 A245011 A245012

Keyword

nonn

Author

Paul D. Hanna, Jul 09 2014

Status

approved