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%I #11 May 08 2019 19:34:15
%S 1,5,16,100,745,5981,54668,542147,5770420,65544681,788168476,
%T 9982471033,132645367515,1842830414090,26692298441517,402114353625235,
%U 6287231891432992,101837297768099079,1705965231481768383,29511920017674005949,526496308467362015150,9674316410154433376601,182882315665489095973391,3552928213442165146349142,70865426835203730805138175
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(3*n) - A(x))^(n+1), where A(0) = 0.
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1+x)^(3*n) - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(3*n*(n-1)) / (1 + x*(1+x)^(3*n)*A(x))^(n+1).
%F (3) 1 = Sum_{n>=0} x^n * (1-x)^(6*n+2) / ((1-x)^(3*n+1) - x*A(x/(1-x)))^(n+1).
%F (4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(3*n-3) * A(x/(1-x)))^n / (1-x)^(3*n^2-2*n-1)).
%e G.f.: A(x) = x + 5*x^2 + 16*x^3 + 100*x^4 + 745*x^5 + 5981*x^6 + 54668*x^7 + 542147*x^8 + 5770420*x^9 + 65544681*x^10 + 788168476*x^11 + 9982471033*x^12 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+x)^3 - A(x))^2 + x^2*((1+x)^6 - A(x))^3 + x^3*((1+x)^9 - A(x))^4 + x^4*((1+x)^12 - A(x))^5 + x^5*((1+x)^15 - A(x))^6 + x^6*((1+x)^18 - A(x))^7 + x^7*((1+x)^21 - A(x))^8 + ...
%e also
%e 1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^3*A(x))^2 + x^2*(1+x)^6/(1 + x*(1+x)^6*A(x))^3 + x^3*(1+x)^18/(1 + x*(1+x)^9*A(x))^4 + x^4*(1+x)^36/(1 + x*(1+x)^12*A(x))^5 + x^5*(1+x)^60/(1 + x*(1+x)^15*A(x))^6 + ...
%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+x +x*O(x^#A))^(3*m) - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A307952, A307954, A307955.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 07 2019
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