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%I #7 Oct 13 2020 03:47:18
%S 1,1,4,104,4196,225216,14845072,1151255440,102289538128,
%T 10226417550096,1135388485042624,138583671424928128,
%U 18446474604149746176,2659732597343823233280,413060592233577210697984,68754628660531280009195776,12213125156726936259944672320,2306358043375070604869802287616,461443265563759624969778550969344,97514484569091438266511351355560448
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} 4^n * ((1+x)^n - A(x))^n.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
%C for any fixed integer k; here, k = 1 with r = 4, p = -A(x), q = (1+x).
%H Paul D. Hanna, <a href="/A326284/b326284.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} 4^n * ((1+x)^n - A(x))^n.
%F (2) 1 = Sum_{n>=0} 4^n * (1+x)^(n^2) / (1 + 4*(1+x)^n*A(x))^(n+1).
%F a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.034391206985341... - _Vaclav Kotesovec_, Oct 13 2020
%e G.f.: A(x) = 1 + x + 4*x^2 + 104*x^3 + 4196*x^4 + 225216*x^5 + 14845072*x^6 + 1151255440*x^7 + 102289538128*x^8 + 10226417550096*x^9 + 1135388485042624*x^10 + ...
%e such that
%e 1 = 1 + 4*((1+x) - A(x)) + 4^2*((1+x)^2 - A(x))^2 + 4^3*((1+x)^3 - A(x))^3 + 4^4*((1+x)^4 - A(x))^4 + 4^5*((1+x)^5 - A(x))^5 + 4^6*((1+x)^6 - A(x))^6 + 4^7*((1+x)^7 - A(x))^7 + ...
%e Also,
%e 1 = 1/(1 + 4*A(x)) + 4*(1+x)/(1 + 4*(1+x)*A(x))^2 + 4^2*(1+x)^4/(1 + 4*(1+x)^2*A(x))^3 + 4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^4 + 4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^5 + 4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^6 + 4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^7 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 4^m*((1+x)^m - Ser(A))^m ) )[#A]/4 ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A303056, A326282, A326283.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 22 2019
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