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A382322 G.f. A(x) satisfies -2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

%I A382322 #13 Mar 22 2025 18:50:12
%S A382322 1,2,8,50,308,2044,14072,100172,730328,5428498,40978780,313322910,
%T A382322 2421454020,18884988540,148443853936,1174814738082,9353539487160,
%U A382322 74865615299260,602057472027484,4862177553583604,39416710563473400,320650120976612168,2616673301770051376,21414973020645504142
%N A382322 G.f. A(x) satisfies -2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%C A382322 Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
%C A382322 Formulas (5) and (6) hold true for all nonzero A(x).
%H A382322 Paul D. Hanna, <a href="/A382322/b382322.txt">Table of n, a(n) for n = 0..301</a>
%F A382322 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.
%F A382322 (1) -2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%F A382322 (2) 2*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
%F A382322 (3) -2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
%F A382322 (4) 2*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
%F A382322 (5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
%F A382322 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).
%e A382322 G.f.: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 308*x^4 + 2044*x^5 + 14072*x^6 + 100172*x^7 + 730328*x^8 + 5428498*x^9 + 40978780*x^10 + ...
%o A382322 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A382322 A[#A] = polcoef(2 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
%o A382322 for(n=0, 30, print1(a(n), ", "))
%Y A382322 Cf. A356783, A380557, A382323.
%K A382322 nonn
%O A382322 0,2
%A A382322 _Paul D. Hanna_, Mar 21 2025