A384273 G.f. A(x) satisfies -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

1, 3, 3, 9, 39, 108, 387, 1581, 5196, 21573, 82596, 318279, 1303146, 5182389, 20919156, 86577264, 351929133, 1462075095, 6077250693, 25277372124, 106131459906, 445859648019, 1878449392365, 7955646845046, 33707865532680, 143344958486019, 610977896794104, 2608218534504888, 11162376089875158

Offset: 0

Author

Paul D. Hanna, Jun 29 2025

Keywords

• nonn

Comments

The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Examples

			G.f.: A(x) = 1 + 3*x + 3*x^2 + 9*x^3 + 39*x^4 + 108*x^5 + 387*x^6 + 1581*x^7 + 5196*x^8 + 21573*x^9 + 82596*x^10 + 318279*x^11 + 1303146*x^12 + ...
where
-3*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A356499, A384271, A384272.

Programs

Mathematica

(* Calculation of constants {d,c}: *) With[{k = 3}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3) * Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r] * QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* Vaclav Kotesovec, Jun 30 2025 *)

PARI

{a(n) = my(A=[1,3]);  for(i=2,n, A=concat(A,0);
A[#A] = polcoef(3*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.

(1) -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

(2) 3*x/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).

(3) -3*x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).

(4) 3*x*theta_4(x)/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).

(5.a) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.

(5.b) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.

(6.a) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.

(6.b) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

a(n) ~ c * d^n / n^(3/2), where d = 4.51389712142676799318649101918382471452597633164586338354... and c = 0.80137914461001211839391539872328025866773400116571811811... - Vaclav Kotesovec, Jun 30 2025