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%I #9 Oct 25 2020 13:18:59
%S 1,3,12,235,7872,351924,19340668,1250971416,92720438955,7733929764167,
%T 716488771114410,72981787493017014,8107675760704948748,
%U 975749719762368998037,126491959992115408069503,17576241581408197850363955,2606439876885873198662077692,410925212330248782377865281826,68641203626673300062880912740755,12110976733338358608040713750036252
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
%H Paul D. Hanna, <a href="/A321603/b321603.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
%F (2) 1 = Sum_{n>=0} (1+x)^(3*n^2) / (1 + (1+x)^(3*n)*A(x))^(n+1).
%e G.f.: A(x) = 1 + 3*x + 12*x^2 + 235*x^3 + 7872*x^4 + 351924*x^5 + 19340668*x^6 + 1250971416*x^7 + 92720438955*x^8 + 7733929764167*x^9 + ...
%e such that
%e 1 = 1 + ((1+x)^3 - A(x)) + ((1+x)^6 - A(x))^2 + ((1+x)^9 - A(x))^3 + ((1+x)^12 - A(x))^4 + ((1+x)^15 - A(x))^5 + ((1+x)^18 - A(x))^6 + ((1+x)^21 - A(x))^7 + ...
%e Also,
%e 1 = 1/(1 + A(x)) + (1+x)^3/(1 + (1+x)^3*A(x))^2 + (1+x)^12/(1 + (1+x)^6*A(x))^3 + (1+x)^27/(1 + (1+x)^9*A(x))^4 + (1+x)^48/(1 + (1+x)^12*A(x))^5 + (1+x)^75/(1 + (1+x)^15*A(x))^6 + ...
%e RELATED SERIES.
%e The logarithmic derivative of the g.f. begins
%e A'(x)/A(x) = 3 + 15*x + 624*x^2 + 28731*x^3 + 1638798*x^4 + 109462350*x^5 + 8333782509*x^6 + 710574703107*x^7 + 67015908514587*x^8 + ...
%e the coefficients of which are all divisible by 3:
%e (1/3) * A'(x)/A(x) = 1 + 5*x + 208*x^2 + 9577*x^3 + 546266*x^4 + 36487450*x^5 + 2777927503*x^6 + 236858234369*x^7 + 22338636171529*x^8 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(3*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A303056, A321602, A321604, A321605.
%Y Cf. A326263.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2018
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