cp's OEIS Frontend

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)).

%I A206637 #10 Mar 30 2012 18:37:35
%S A206637 1,2,10,50,266,1466,8370,49090,294458,1798794,11156074,70069514,
%T A206637 444822530,2849764698,18401517066,119640989514,782575127258,
%U A206637 5146252178882,34003440381186,225635772455882,1503017848153914,10046960505610082,67372689978768714,453099298491559554
%N 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)).
%F A206637 G.f. satisfies the identities:
%F A206637 (1) A(x) = 1 + Sum_{n>=1} 2*x^n*A(x)^n / Product_{k=1..n} (1 - 2*x^k).
%F A206637 (2) A(x) = 1 + Sum_{n>=1} 2^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).
%e A206637 G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^4 + 1466*x^5 + 8370*x^6 +...
%e A206637 where the g.f. satisfies:
%e A206637 (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))) +...
%e A206637 (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)) +...
%e A206637 (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))) +...
%o A206637 (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)}
%o A206637 (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)}
%o A206637 (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)}
%o A206637 for(n=0,35,print1(a(n),","))
%Y A206637 Cf. A145268, A206638.
%K A206637 nonn
%O A206637 0,2
%A A206637 _Paul D. Hanna_, Feb 10 2012