%I #7 Jan 23 2024 08:41:14

%S 1,2,2,12,22,144,314,2094,4980,33962,85104,589050,1526520,10690958,

%T 28357414,200513838,540848844,3855418084,10529575602,75589847732,

%U 208400923746,1505463570864,4180703306104,30372614867534,84817986727644,619420989037864,1737266065061962

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

%H Paul D. Hanna, <a href="/A369086/b369086.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.

%F (2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.

%F (3) 2 = Product_{n>=1} (1 + (-x)^(n-1)*A(x)) * (1 + (-x)^n/A(x)) * (1 - (-x)^n), by the Jacobi Triple Product identity.

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 12*x^3 + 22*x^4 + 144*x^5 + 314*x^6 + 2094*x^7 + 4980*x^8 + 33962*x^9 + 85104*x^10 + 589050*x^11 + 1526520*x^12 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

%o A[#A] = polcoeff(2 - sum(m=-#A, #A, (-x)^(m*(m-1)/2) * Ser(A)^m ), #A-1)); H=A; A[n+1]}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A354645.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 22 2024