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A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)).

%I A206638 #9 Feb 20 2025 14:04:50
%S A206638 1,3,21,147,1074,8076,62454,494292,3990378,32756142,272715870,
%T A206638 2297982828,19563641319,168036314862,1454458825605,12674387617266,
%U A206638 111104771086812,979101922849230,8668964794053837,77080072176742422,687976906966730076,6161811541538326680
%N A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)).
%F A206638 G.f. satisfies the identities:
%F A206638 (1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k).
%F A206638 (2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).
%e A206638 G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +...
%e A206638 where the g.f. satisfies:
%e A206638 (0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
%e A206638 (1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +...
%e A206638 (2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
%o A206638 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
%o A206638 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)}
%o A206638 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
%o A206638 for(n=0,35,print1(a(n),", "))
%Y A206638 Cf. A145268, A196150, A206637.
%K A206638 nonn
%O A206638 0,2
%A A206638 _Paul D. Hanna_, Feb 10 2012