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%I #12 May 16 2019 09:56:36
%S 1,5,10,80,568,4220,38692,369602,3829789,42483419,498335248,
%T 6168187340,80190252964,1090909725218,15487454931220,228882342189464,
%U 3513421961681770,55912182446264327,920864428915749175,15671937126462121502,275216319427229910676,4980676147299194153192,92778491004412178347075,1776939414715404683846648,34955882406696210297175882,705630056440779526097189330
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 3*x)^n - A(x))^(n+1), where A(0) = 0.
%H Paul D. Hanna, <a href="/A325583/b325583.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1 + 3*x)^n - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1 + 3*x)^(n*(n-1)) / (1 + x*(1 + 3*x)^n*A(x))^(n+1).
%F FORMULA FOR TERMS.
%F a(n) = (-1)^n (mod 3) for n >= 0.
%e G.f.: A(x) = x + 5*x^2 + 10*x^3 + 80*x^4 + 568*x^5 + 4220*x^6 + 38692*x^7 + 369602*x^8 + 3829789*x^9 + 42483419*x^10 + 498335248*x^11 + 6168187340*x^12 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+3*x) - A(x))^2 + x^2*((1+3*x)^2 - A(x))^3 + x^3*((1+3*x)^3 - A(x))^4 + x^4*((1+3*x)^4 - A(x))^5 + x^5*((1+3*x)^5 - A(x))^6 + x^6*((1+3*x)^6 - A(x))^7 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1 + 3*x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A325582, A325584, A325585.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 11 2019
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