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A361766 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n/A(-x))^(n+2).

%I A361766 #15 Jan 21 2025 22:20:28
%S A361766 1,1,2,5,12,27,57,123,280,666,1614,3955,9733,23949,58967,145844,
%T A361766 363137,910339,2295192,5811070,14754567,37542078,95715596,244567665,
%U A361766 626388406,1608131393,4137707994,10667045757,27546269363,71241831762,184508259405,478501423792
%N A361766 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n/A(-x))^(n+2).
%C A361766 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.
%C A361766 Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n) * (y - x^n)^(n+2), which holds for all y as a formal power series in x. - _Paul D. Hanna_, Jan 19 2025
%H A361766 Paul D. Hanna, <a href="/A361766/b361766.txt">Table of n, a(n) for n = 0..500</a>
%F A361766 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A361766 (1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) - (-x)^n)^(n+2) / A(x)^n.
%F A361766 (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) * A(x)^n / (1 - (-x)^n*A(x))^(n-2).
%F A361766 From _Paul D. Hanna_, Jan 19 2025: (Start)
%F A361766 (3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (1 - (-x)^n)^n * A(x)^n.
%F A361766 (4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) / ((1 - (-x)^n)^n * A(x)^n).
%F A361766 (End)
%F A361766 a(n) ~ c * d^n / n^(3/2), where d = 2.71312501383... and c = 3.43853109... - _Vaclav Kotesovec_, Mar 31 2023
%e A361766 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...
%o A361766 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A361766 A[#A] = -polcoeff( sum(m=-#A,#A, (-x)^m * (1 - (-x)^m/Ser(A))^(m+2) ), #A-3));A[n+1]}
%o A361766 for(n=0,35,print1(a(n),", "))
%Y A361766 Cf. A358952, A355866.
%K A361766 nonn
%O A361766 0,3
%A A361766 _Paul D. Hanna_, Mar 26 2023