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A304860 G.f. A(x) satisfies: x = Sum_{n>=0} ( (1+x)^(n^2) - A(x)^n ) / 2^(n+1).

%I A304860 #12 May 28 2018 12:07:22
%S A304860 1,2,32,608,17750,683504,32183336,1782735248,113381031512,
%T A304860 8138225237204,650735042088080,57369033007665680,5529284312514428840,
%U A304860 578479328396134930928,65297339893598788494368,7910610591246432715704704,1023854667471171305890388408,141001918216059025744295715872,20587944237516075824024078357264,3176963079503660078673757802123360
%N A304860 G.f. A(x) satisfies: x = Sum_{n>=0} ( (1+x)^(n^2) - A(x)^n ) / 2^(n+1).
%H A304860 Paul D. Hanna, <a href="/A304860/b304860.txt">Table of n, a(n) for n = 0..60</a>
%F A304860 G.f. A(x) satisfies:
%F A304860 (1) x = Sum_{n>=0} ( (1+x)^(n^2) - A(x)^n ) / 2^(n+1).
%F A304860 (2) A(x) = 2 - 1/(G(x) - x), where G(x) = Sum_{n>=0} (1+x)^(n^2) / 2^(n+1) is the g.f. of A173217.
%e A304860 G.f.: A(x) = 1 + 2*x + 32*x^2 + 608*x^3 + 17750*x^4 + 683504*x^5 + 32183336*x^6 + 1782735248*x^7 + 113381031512*x^8 + 8138225237204*x^9 + ...
%e A304860 such that
%e A304860 x = ((1+x) - A(x))/2^2 + ((1+x)^4 - A(x)^2)/2^3 + ((1+x)^9 - A(x)^3)/2^4 + ((1+x)^16 - A(x)^4)/2^5 + ((1+x)^25 - A(x)^5)/2^6 + ((1+x)^36 - A(x)^6)/2^7 + ...
%e A304860 RELATED SERIES.
%e A304860 G(x) = Sum_{n>=0} (1+x)^(n^2) / 2^(n+1) = 1 + 3*x + 36*x^2 + 744*x^3 + 21606*x^4 + 807912*x^5 + 36948912*x^6 + 1997801520*x^7 + 124666314300*x^8 + ... + A173217(n)*x^n + ...
%e A304860 1/(2 - A(x)) = G(x) - x = 1 + 2*x + 36*x^2 + 744*x^3 + 21606*x^4 + 807912*x^5 + 36948912*x^6 + 1997801520*x^7 + 124666314300*x^8 + ...
%e A304860 Let F(x) satisfy
%e A304860 x = Sum_{n>=0} ( F(x)^n - A(x)^n ) / 2^(n+1), then
%e A304860 F(x) = 1 + 3*x + 27*x^2 + 555*x^3 + 16737*x^4 + 652815*x^5 + 30967917*x^6 + 1724292411*x^7 + 110091861729*x^8 + 7926482395935*x^9 + ...
%e A304860 where 1/(2 - F(x)) = x + 1/(2 - A(x)).
%Y A304860 Cf. A173217.
%K A304860 nonn
%O A304860 0,2
%A A304860 _Paul D. Hanna_, May 28 2018