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%I #8 Oct 01 2019 02:57:44
%S 1,2,25,526,15554,590575,27338911,1492079652,93776123520,
%T 6669479907548,529549414923174,46432923079132507,4456442206987048919,
%U 464691980829431905959,52313905571976063712791,6323967971227649063422416,817029035199692037187367976,112348605647113658601511442450,16383091093205410144470895801204,2525293749572595915714422633867944
%N G.f.: A(x) = Sum_{n>=0} ((1+x)^n - x*A(x))^n / 2^(n+1).
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} ((1+x)^n - x*A(x))^n / 2^(n+1).
%F (2) A(x) = Sum_{n>=0} (1+x)^(n^2) / (2 + x*A(x))^(n+1).
%e G.f.: A(x) = 1 + 2*x + 25*x^2 + 526*x^3 + 15554*x^4 + 590575*x^5 + 27338911*x^6 + 1492079652*x^7 + 93776123520*x^8 + 6669479907548*x^9 + ...
%e such that
%e A(x) = 1/2 + ((1+x) - x*A(x))/2^2 + ((1+x)^2 - x*A(x))^2/2^3 + ((1+x)^3 - x*A(x))^3/2^4 + ((1+x)^4 - x*A(x))^4/2^5 + ((1+x)^5 - x*A(x))^5/2^6 + ...
%e also,
%e A(x) = 1/(2 + x*A(x)) + (1+x)/(2 + x*A(x))^2 + (1+x)^4/(2 + x*A(x))^3 + (1+x)^9/(2 + x*A(x))^4 + (1+x)^16/(2 + x*A(x))^5 + (1+x)^25/(2 + x*A(x))^6 + ...
%t m = 20; A[_] = 0;
%t Do[A[x_] = Sum[((1+x)^n - x A[x])^n/2^(n+1), {n, 0, k^2}] + O[x]^k, {k, 1, m}];
%t CoefficientList[A[x], x] // Round (* _Jean-François Alcover_, Oct 01 2019 *)
%Y Cf. A322728.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 31 2019
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