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A354645 G.f. A(x) satisfies: -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.

%I A354645 #12 Jan 23 2024 08:41:10
%S A354645 1,1,3,9,23,62,179,571,1888,6309,21114,71387,245162,854703,3010602,
%T A354645 10673814,38036229,136250135,490696565,1776089734,6456234450,
%U A354645 23554296293,86210280655,316481218498,1165089759958,4300364442451,15910717070706,58996259565000
%N A354645 G.f. A(x) satisfies: -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
%H A354645 Paul D. Hanna, <a href="/A354645/b354645.txt">Table of n, a(n) for n = 0..500</a>
%F A354645 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A354645 (1) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
%F A354645 (2) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A354645 (3) x = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (A(x)^(n+1) - 1/A(x)^n).
%F A354645 (4) -x = Product_{n>=1} (1 - x^(n-1)*A(x)) * (1 - x^n/A(x)) * (1 - x^n), by the Jacobi Triple Product identity. - _Paul D. Hanna_, Jan 22 2024
%e A354645 G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 23*x^4 + 62*x^5 + 179*x^6 + 571*x^7 + 1888*x^8 + 6309*x^9 + 21114*x^10 + 71387*x^11 + 245162*x^12 + ...
%e A354645 such that
%e A354645 x = (A(x) - 1) - x*(A(x)^2 - 1/A(x)) + x^3*(A(x)^3 - 1/A(x)^2) - x^6*(A(x)^4 - 1/A(x)^3) + x^10*(A(x)^5 - 1/A(x)^4) - x^15*(A(x)^6 - 1/A(x)^5) + x^21*(A(x)^7 - 1/A(x)^6) + ...
%e A354645 Also, we have the infinite product
%e A354645 -x = (1 - A(x))*(1 - x/A(x))*(1-x) * (1 - x*A(x))*(1 - x^2/A(x))*(1-x^2) * (1 - x^2*A(x))*(1 - x^3/A(x))*(1-x^3) * (1 - x^3*A(x))*(1 - x^4/A(x))*(1-x^4) * ...
%o A354645 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A354645 A[#A] = polcoeff(x + sum(m=-#A,#A, (-1)^m * x^(m*(m-1)/2) * Ser(A)^m ),#A-1));H=A;A[n+1]}
%o A354645 for(n=0,40,print1(a(n),", "))
%Y A354645 Cf. A369086.
%K A354645 nonn
%O A354645 0,3
%A A354645 _Paul D. Hanna_, Jun 07 2022