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A363143 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^(n+1).

%I A363143 #13 May 18 2023 19:22:06
%S A363143 1,1,1,3,7,13,25,52,111,235,495,1054,2271,4923,10703,23354,51190,
%T A363143 112668,248783,550875,1223107,2722766,6075619,13586390,30442616,
%U A363143 68339788,153683822,346173172,780948750,1764312745,3991321375,9040912764,20503640896,46552634034
%N A363143 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^(n+1).
%H A363143 Paul D. Hanna, <a href="/A363143/b363143.txt">Table of n, a(n) for n = 0..300</a>
%F A363143 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A363143 (1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^(n+1).
%F A363143 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 + A(x)*x^(3*n+1))^(n-1).
%F A363143 (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^n.
%F A363143 (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (A(x) + x^(3*n-1))^(n-1).
%F A363143 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n^2) / (1 + A(x)*x^(3*n+1))^n.
%e A363143 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 25*x^6 + 52*x^7 + 111*x^8 + 235*x^9 + 495*x^10 + 1054*x^11 + 2271*x^12 + ...
%o A363143 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A363143 A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(3*m-1))^(m+1) ),#A-1));A[n+1]}
%o A363143 for(n=0,40,print1(a(n),", "))
%Y A363143 Cf. A359711, A363142, A363144.
%K A363143 nonn
%O A363143 0,4
%A A363143 _Paul D. Hanna_, May 17 2023