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%I #7 Apr 06 2018 20:37:04
%S 1,1,8,112,1972,39404,853892,19591692,469250416,11628163256,
%T 296351290004,7736140181364,206273152705660,5606990999026252,
%U 155184267041459384,4370129283473065984,125189806731347999476,3648813481714933367516,108265665575110494127284,3273367006162760350945260,100977120404026793376264880,3183255539561434435490787720
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (3 + x*A(x)^n)^n / 4^(n+1).
%C Compare to: G(x) = Sum_{n>=0} (3 + x*G(x)^k)^n / 4^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.
%H Paul D. Hanna, <a href="/A302104/b302104.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} (3 + x*A(x)^n)^n / 4^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (4 - 3*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 8*x^2 + 112*x^3 + 1972*x^4 + 39404*x^5 + 853892*x^6 + 19591692*x^7 + 469250416*x^8 + 11628163256*x^9 + 296351290004*x^10 + ...
%e such that
%e A(x) = 3/4 + (3 + x*A(x))/4^2 + (3 + x*A(x)^2)^2/4^3 + (3 + x*A(x)^3)^3/4^4 + (3 + x*A(x)^4)^4/4^5 + (3 + x*A(x)^5)^5/4^6 + (3 + x*A(x)^6)^6/4^7 + ...
%e Also, due to a series identity,
%e A(x) = 1 + x*A(x)/(4 - 3*A(x))^2 + x^2*A(x)^4/(4 - 3*A(x)^2)^3 + x^3*A(x)^9/(4 - 3*A(x)^3)^4 + x^4*A(x)^16/(4 - 3*A(x)^4)^5 + x^5*A(x)^25/(4 - 3*A(x)^5)^6 + x^6*A(x)^36/(4 - 3*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (4 - 3*A(x)^n)^(n+1) + ...
%o (PARI) {a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (4 - 3*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A300050, A302103, A302105.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2018
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