A369682 Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = theta_2(x^(1/2)) / x^(1/8).

1, 4, 12, 38, 112, 332, 972, 2818, 8098, 23096, 65418, 184194, 516080, 1440334, 4008442, 11135682, 30912896, 85835538, 238601354, 664447912, 1854592214, 5189848462, 14561237108, 40954656118, 115428662380, 325847049200, 920772219740, 2602948470362, 7356994944096, 20779322594048

Offset: 0

Author

Paul D. Hanna, Feb 04 2024

Keywords

• nonn

Comments

Note: theta_2(x^(1/2)) / x^(1/8) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) - see A089799.

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 38*x^3 + 112*x^4 + 332*x^5 + 972*x^6 + 2818*x^7 + 8098*x^8 + 23096*x^9 + 65418*x^10 + 184194*x^11 + 516080*x^12 + ...
By definition, A = A(x) satisfies the sum of products
theta_2(x^(1/2))/x^(1/8) = (1 + A) - x*(1 + A)*(x + A) + x^2*(1 + A)*(x + A)*(x^2 + A) - x^3*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A) + x^4*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) -+ ...
also, A = A(x) satisfies another sum of products
x*theta_2(x^(1/2))/x^(1/8) = 1 - 1/(1 + x*A) + x/((1 + x*A)*(1 + x^2*A)) - x^3/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)) + x^6/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)) - x^10/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)*(1 + x^5*A)) +- ...
Further, A = A(x) satisfies the continued fraction given by
theta_2(x^(1/2))/x^(1/8) = (1 + A)/(1 + x*(x + A)/(1 - x*(x + A) + x*(x^2 + A)/(1 - x*(x^2 + A) + x*(x^3 + A)/(1 - x*(x^3 + A) + x*(x^4 + A)/(1 - x*(x^4 + A) + x*(x^5 + A)/(1 - x*(x^5 + A) + ...))))))
where theta_2(x^(1/2))/x^(1/8) = 2 + 2*x + 2*x^3 + 2*x^6 + 2*x^10 + 2*x^15 + 2*x^21 + ... + 2*x^(n*(n+1)/2) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..325
• Vaclav Kotesovec, Plot of a(n+1)/a(n) for n = 1..324

Crossrefs

Cf. A369683, A369684, A089799 (theta_2).

Programs

PARI

{a(n) = my(A=[1], M = sqrtint(2*n)+1); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-M,M, x^(n*(n+1)/2) ) - sum(n=0,#A, (-1)^n * x^n * prod(k=0,n, x^k + Ser(A)) ), #A-1) ); H=A; 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.

(1) Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = Sum_{n=-oo..+oo} x^(n*(n+1)/2).

(2) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) / Product_{k=1..n} (1 + x^k*A(x)) = x * Sum_{n=-oo..+oo} x^(n*(n+1)/2).

(3) theta_2(x^(1/2))/x^(1/8) = (1 + A(x))/(1 + F(1)), where F(n) = x*(x^n + A(x))/(1 - x*(x^n + A(x)) + F(n+1)), a continued fraction.

(4) x * theta_2(x^(1/2))/x^(1/8) = 1/(1 + F(1)), where F(n) = x^(n-1)/(1 - x^(n-1) + x^n*A + (1 + x^n*A) * F(n+1)), a continued fraction.