A245009 - OEIS

%I #10 Nov 27 2017 18:12:57

%S 1,2,10,88,978,12200,163156,2286448,33138874,492657384,7470940300,

%T 115115319376,1797128902132,28364816229008,451870965523368,

%U 7256283996155360,117333885356923274,1908844190372949224,31221135850863938268,513100005743085437328,8468653781083527106012,140314257925457275837488

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

%F G.f. satisfies:

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

%F (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).

%F (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.

%F 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

%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 88*x^3 + 978*x^4 + 12200*x^5 +...

%e Compare A(x)^4 to 8*(1-x)*A(x):

%e A(x)^4 = 1 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...

%e 8*(1-x)*A(x) = 8 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...

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

%o (PARI) {a(n)=polcoeff(1 + serreverse((1+8*x - (1+x)^4)/(8*(1+x +x*O(x^n)))), n)}

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

%o (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]}

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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 09 2014