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A363144 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).

%I A363144 #13 May 18 2023 19:23:18
%S A363144 1,1,1,1,3,7,13,21,35,64,125,243,459,852,1593,3035,5857,11326,21835,
%T A363144 42053,81246,157741,307421,600207,1172805,2294197,4495735,8827574,
%U A363144 17363422,34198201,67429181,133097669,263028031,520406201,1030749582,2043553947,4055171751
%N A363144 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).
%H A363144 Paul D. Hanna, <a href="/A363144/b363144.txt">Table of n, a(n) for n = 0..300</a>
%F A363144 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A363144 (1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).
%F A363144 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(4*n*(n-1)) / (1 + A(x)*x^(4*n+1))^(n-1).
%F A363144 (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^n.
%F A363144 (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(5*n) * (A(x) + x^(4*n-1))^(n-1).
%F A363144 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(4*n^2) / (1 + A(x)*x^(4*n+1))^n.
%e A363144 G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 + 35*x^8 + 64*x^9 + 125*x^10 + 243*x^11 + 459*x^12 + ...
%o A363144 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A363144 A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(4*m-1))^(m+1) ),#A-1));A[n+1]}
%o A363144 for(n=0,40,print1(a(n),", "))
%Y A363144 Cf. A359711, A363142, A363143.
%K A363144 nonn
%O A363144 0,5
%A A363144 _Paul D. Hanna_, May 17 2023