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%I A302103 #8 Apr 06 2018 20:36:36
%S A302103 1,1,6,63,837,12672,208686,3647568,66697203,1264307667,24696153573,
%T A302103 495076265421,10157438738790,212900154037875,4553735135491134,
%U A302103 99341289091151409,2210262851488661562,50173932628981325523,1162965513498859292415,27554435907912281877315,668277970101220006626558,16617278354076763108026795
%N A302103 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).
%C A302103 Compare to: G(x) = Sum_{n>=0} (2 + x*G(x)^k)^n / 3^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.
%H A302103 Paul D. Hanna, <a href="/A302103/b302103.txt">Table of n, a(n) for n = 0..100</a>
%F A302103 G.f. A(x) satisfies:
%F A302103 (1) A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).
%F A302103 (2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1).
%e A302103 G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 837*x^4 + 12672*x^5 + 208686*x^6 + 3647568*x^7 + 66697203*x^8 + 1264307667*x^9 + 24696153573*x^10 + ...
%e A302103 such that
%e A302103 A(x) = 2/3 + (2 + x*A(x))/3^2 + (2 + x*A(x)^2)^2/3^3 + (2 + x*A(x)^3)^3/3^4 + (2 + x*A(x)^4)^4/3^5 + (2 + x*A(x)^5)^5/3^6 + (2 + x*A(x)^6)^6/3^7 + ...
%e A302103 Also, due to a series identity,
%e A302103 A(x) = 1 + x*A(x)/(3 - 2*A(x))^2 + x^2*A(x)^4/(3 - 2*A(x)^2)^3 + x^3*A(x)^9/(3 - 2*A(x)^3)^4 + x^4*A(x)^16/(3 - 2*A(x)^4)^5 + x^5*A(x)^25/(3 - 2*A(x)^5)^6 + x^6*A(x)^36/(3 - 2*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1) + ...
%o A302103 (PARI) {a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (3 - 2*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
%o A302103 for(n=0,30,print1(a(n),", "))
%Y A302103 Cf. A300050, A302104, A302105.
%K A302103 nonn
%O A302103 0,3
%A A302103 _Paul D. Hanna_, Apr 05 2018

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