cp's OEIS Frontend

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

%I A244594 #16 Jul 10 2014 15:05:16
%S A244594 1,1,4,29,263,2672,29088,331749,3912660,47329811,583983656,7321173872,
%T A244594 92990672635,1194113490556,15476763809428,202197552311829,
%U A244594 2659975668005367,35205831900984144,468468683002725372,6263539340729569047,84103985900174324256,1133671250214654009000
%N A244594 G.f. satisfies: A(x) = (3 + A(x)^3) / (4 - x).
%H A244594 Vaclav Kotesovec, <a href="/A244594/b244594.txt">Table of n, a(n) for n = 0..400</a>
%F A244594 G.f. satisfies:
%F A244594 (1) A(x) = 1 + Series_Reversion( (1+4*x - (1+x)^3)/(1 + x) ).
%F A244594 (2) A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (3 + x*A(x))^(2*n+1) / 4^(3*n+1).
%F A244594 (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.
%F A244594 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
%F A244594 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
%e A244594 G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 263*x^4 + 2672*x^5 + 29088*x^6 +...
%e A244594 Compare
%e A244594 (4 - x)*A(x) = 4 + 3*x + 15*x^2 + 112*x^3 + 1023*x^4 + 10425*x^5 +...
%e A244594 to:
%e A244594 A(x)^3 = 1 + 3*x + 15*x^2 + 112*x^3 + 1023*x^4 + 10425*x^5 + 113680*x^6 +...
%o A244594 (PARI) {a(n)=polcoeff(1+serreverse((1+4*x - (1+x)^3)/(1 + x +x*O(x^n))),n)}
%o A244594 for(n=0,30,print1(a(n),", "))
%o A244594 (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]}
%o A244594 for(n=0,30,print1(a(n),", "))
%Y A244594 Cf. A120590, A001764.
%K A244594 nonn
%O A244594 0,3
%A A244594 _Paul D. Hanna_, Jul 01 2014