A244627 - OEIS

%I #7 Jul 03 2014 03:45:11

%S 1,2,10,84,882,10380,130916,1729960,23640770,331357276,4737405356,

%T 68818101400,1012852747220,15070913484664,226340757825800,

%U 3426481380787024,52232521742541410,801068351764540540,12351730365539402076,191363830520437179960

%N G.f. satisfies: A(x) = (4 + A(x)^3) / (5 - 4*x).

%H Vaclav Kotesovec, <a href="/A244627/b244627.txt">Table of n, a(n) for n = 0..350</a>

%F G.f. satisfies:

%F (1) A(x) = 1 + 2*Series_Reversion( x*(1 - 3*x - 2*x^2)/(1 + 2*x) ).

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

%F (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 A120592.

%F Recurrence: 17*(n-1)*n*a(n) = 150*(n-1)*(2*n-3)*a(n-1) - 40*(6*n^2 - 24*n + 23)*a(n-2) + 32*(n-2)*(2*n-7)*a(n-3). - _Vaclav Kotesovec_, Jul 03 2014

%F a(n) ~ sqrt(15-9*2^(2/3)) * ((60*2^(2/3)+72*2^(1/3)+100)/17)^n / (6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 03 2014

%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 84*x^3 + 882*x^4 + 10380*x^5 + 130916*x^6 +...

%e Compare

%e (5 - 4*x)*A(x) = 5 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 +...

%e to:

%e A(x)^3 = 1 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 + 613060*x^6 +...

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

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

%o (PARI) {a(n)=local(A=[1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^3 - (5-4*x)*Ax )/2 )[#A]); A[n+1]}

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

%Y Cf. A120592, A244594.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 02 2014