A384272 G.f. A(x) satisfies -2*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

1, 2, 2, 6, 16, 50, 144, 478, 1510, 5116, 17034, 58812, 202166, 709228, 2489546, 8848146, 31525526, 113236920, 407983964, 1478249454, 5372468156, 19607233026, 71758722172, 263480958508, 969856453650, 3579426292768, 13239549874552, 49078409375334, 182282423994240, 678289439131812, 2528257204808848

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 + 2*x + 2*x^2 + 6*x^3 + 16*x^4 + 50*x^5 + 144*x^6 + 478*x^7 + 1510*x^8 + 5116*x^9 + 17034*x^10 + 58812*x^11 + 202166*x^12 + ...
where
-2*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, A384273.

Programs

Mathematica

(* Calculation of constants {d,c}: *) With[{k = 2}, 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=[1,2]);  for(i=2,n, A=concat(A,0);
A[#A] = polcoef(2*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) -2*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

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

(3) -2*x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).

(4) 2*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) -2*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.

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

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

(6.b) 2*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 = 3.9182818074503417233561248171647191927022193746074095378101... and c = 0.687770752477136312107316168146720576083024421405682875987... - Vaclav Kotesovec, Jun 30 2025