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%I #7 Apr 06 2018 20:37:26
%S 1,1,10,175,3835,95090,2551480,72360700,2139052845,65329175385,
%T 2049247480265,65752776679275,2151923601749290,71691421965972905,
%U 2428004656549037580,83523871228996755395,2917260885363111908770,103451501815230690971935,3726040763307222530311125,136400452641372633368206185,5080478361492407723101242440
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).
%C Compare to: G(x) = Sum_{n>=0} (4 + x*G(x)^k)^n / 5^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.
%H Paul D. Hanna, <a href="/A302105/b302105.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (5 - 4*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 10*x^2 + 175*x^3 + 3835*x^4 + 95090*x^5 + 2551480*x^6 + 72360700*x^7 + 2139052845*x^8 + 65329175385*x^9 + 2049247480265*x^10 + ...
%e such that
%e A(x) = 4/5 + (4 + x*A(x))/5^2 + (4 + x*A(x)^2)^2/5^3 + (4 + x*A(x)^3)^3/5^4 + (4 + x*A(x)^4)^4/5^5 + (4 + x*A(x)^5)^5/5^6 + (4 + x*A(x)^6)^6/5^7 + ...
%e Also, due to a series identity,
%e A(x) = 1 + x*A(x)/(5 - 4*A(x))^2 + x^2*A(x)^4/(5 - 4*A(x)^2)^3 + x^3*A(x)^9/(5 - 4*A(x)^3)^4 + x^4*A(x)^16/(5 - 4*A(x)^4)^5 + x^5*A(x)^25/(5 - 4*A(x)^5)^6 + x^6*A(x)^36/(5 - 4*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (5 - 4*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) / (5 - 4*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A300050, A302103, A302104.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2018
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