A206637 G.f. satisfies: A(x) = Sum_{n>=0} 2^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 2*x^k)*(1 - x^k*A(x)).

1, 2, 10, 50, 266, 1466, 8370, 49090, 294458, 1798794, 11156074, 70069514, 444822530, 2849764698, 18401517066, 119640989514, 782575127258, 5146252178882, 34003440381186, 225635772455882, 1503017848153914, 10046960505610082, 67372689978768714, 453099298491559554

Offset: 0

Author

Paul D. Hanna, Feb 10 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^4 + 1466*x^5 + 8370*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + 2*x*A(x)/((1-2*x)*(1-x*A(x))) + 4*x^4*A(x)^2/((1-2*x)*(1-2*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 8*x^9*A(x)^3/((1-2*x)*(1-2*x^2)*(1-2*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(1) A(x) = 1 + 2*x*A(x)/(1-2*x) + 2*x^2*A(x)^2/((1-2*x)*(1-2*x^2)) + 2*x^3*A(x)^3/((1-2*x)*(1-2*x^2)*(1-2*x^3)) +...
(2) A(x) = 1 + 2*x*A(x)/(1-x*A(x)) + 4*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 8*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
		

Crossrefs

Cf. A145268, A206638.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*2^m*A^m/prod(k=1, m, (1-2*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)}

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 2*x^m*A^m/prod(k=1, m, (1-2*x^k+x*O(x^n))))); polcoeff(A, n)}

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 2^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),","))

Formula

G.f. satisfies the identities:

(1) A(x) = 1 + Sum_{n>=1} 2*x^n*A(x)^n / Product_{k=1..n} (1 - 2*x^k).

(2) A(x) = 1 + Sum_{n>=1} 2^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).