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a(n) ~ 2^(n/2 - 2) * 3^(3*(n-1)/4) / (sqrt(Pi) * n^(3/2) * (5*sqrt(2)*3^(3/4) - 16)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017

CoefficientList[1 + InverseSeries[Series[(1+5*x - (1+x)^4)/(1+x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)

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G.f. satisfies: A(x) = (4 + A(x)^4) / (5-x).

G.f. satisfies:

(1) A(x) = 1 + Series_Reversion( (1+5*x* - (1 - 6*+x - )^4*x^2 - x^3)/(1 + x) ).

G.f.: A(x) = 1 + x + 7*x^2 + 95*x^3 + 1614*x^4 + 30718*x^5 +...

(PARI) {a(n)=polcoeff(1 + serreverse((1+5*x* - (1 - 6*+x - )^4*x^2 - x^3)/(1 + x +x*O(x^n))), n)}

Cf. A120593, A244627, A244594.

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allocated for Paul D. Hanna

G.f. satisfies: A(x) = (4 + A(x)^4) / (5-x).

1, 1, 7, 95, 1614, 30718, 626434, 13383650, 295692145, 6700461777, 154871912815, 3637093846055, 86539594779772, 2081721640140460, 50542732376144460, 1236960716959913020, 30483096737455969766, 755783491624380578998, 18839297079646725396450

0,3

G.f. satisfies:

(1) A(x) = 1 + Series_Reversion( x*(1 - 6*x - 4*x^2 - x^3)/(1 + x) ).

(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4 + x*A(x))^(3*n+1) / 5^(4*n+1).

(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (4+x + G(x)^4)/5 is the g.f. of A120593.

G.f.: A(x) = 1 + x + 7*x^2 + 95*x^3 + 1614*x^4 + 30718*x^5 +...

Compare A(x)^4 to (5-x)*A(x):

A(x)^4 = 1 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...

(5-x)*A(x) = 5 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...

(PARI) {a(n)=polcoeff(1 + serreverse(x*(1 - 6*x - 4*x^2 - x^3)/(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 - (5-x)*Ax ) )[#A]); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

Cf. A120593, A244627, A244594.

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nonn

Paul D. Hanna, Jul 09 2014

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