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%I #17 Sep 30 2023 05:24:40
%S 1,4,14,84,444,2928,18214,125428,844534,5989816,42186878,305757288,
%T 2215509018,16326672796,120612763510,900561207232,6746557569136,
%U 50906726784700,385432963013140,2933390906035044,22395805754363208,171660252748284852,1319474586701337644
%N G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).
%C Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0.
%H Paul D. Hanna, <a href="/A356508/b356508.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) satisfies:
%F (1) 2/P(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n, where P(x) = 1/Product_{n>=1} (1 - x^n) is the partition function (A000041)..
%F (2) 2 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), by the Jacobi triple product identity.
%F a(n) ~ c * d^n / n^(3/2), where d = 8.2221649228195625... and c = 1.06682907735826... - _Vaclav Kotesovec_, Mar 14 2023
%e G.f.: A(x) = 1 + 4*x + 14*x^2 + 84*x^3 + 444*x^4 + 2928*x^5 + 18214*x^6 + 125428*x^7 + 844534*x^8 + 5989816*x^9 + 42186878*x^10 + ...
%e such that
%e 2 = (1 + x*A(x))*(1 + 1/A(x)) * (1 + x^2*A(x))*(1 + x/A(x)) * (1 + x^3*A(x))*(1 + x^2/A(x)) * (1 + x^4*A(x))*(1 + x^3/A(x)) * (1 + x^5*A(x))*(1 + x^4/A(x)) * ...
%e also,
%e 2/P(x) = ... + x^10/A(x)^5 + x^6/A(x)^4 + x^3/A(x)^3 + x/A(x)^2 + 1/A(x) + 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 + ... + x^(n*(n+1)/2) * A(x)^n + ...
%e where P(x) is the partition function and begins
%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
%e and
%e 2/P(x) = 2 - 2*x - 2*x^2 + 2*x^5 + 2*x^7 - 2*x^12 - 2*x^15 + 2*x^22 + 2*x^26 - 2*x^35 - 2*x^40 + 2*x^51 + 2*x^57 - 2*x^70 - 2*x^77 + 2*x^92 + 2*x^100 + ...
%t (* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2 * Sqrt[-(r*(1 + s)*(QPochhammer[-s, r]^2* Derivative[0, 1][QPochhammer][-1/s, r] + 2*(1 + s) * Derivative[0, 1][QPochhammer][-s, r])) / (Pi * QPochhammer[-s, r] * (s* Log[r]^2 + (1 + s)^2*(QPolyGamma[1, Log[-1/s]/Log[r], r] + QPolyGamma[1, Log[-s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1/s, r]*QPochhammer[-s, r] == 2*(1 + s), (1 + s)*(QPolyGamma[0, Log[-1/s]/Log[r], r] - QPolyGamma[0, Log[-s]/Log[r], r]) == s*Log[r]}, {r, 1/8}, {s, 3}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Sep 30 2023 *)
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff( -2 + prod(n=1, #A, (1 + x^n*Ser(A)) * (1 + x^(n-1)/Ser(A)) ), #A-1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1], M, P=prod(k=1, n, 1-x^k +x*O(x^n))); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*n+9));
%o A[#A] = polcoeff( -2*P + sum(m=-M, M, x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A356499, A000041, A052002.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 11 2022
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