A369683 Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = theta_3(x).

1, 2, 4, 8, 19, 40, 86, 181, 383, 811, 1709, 3598, 7554, 15839, 33158, 69319, 144731, 301813, 628727, 1308487, 2720908, 5653743, 11740260, 24365703, 50544832, 104810967, 217270721, 450287996, 933043086, 1933125817, 4004865745, 8296690701, 17188106646, 35609996584

Offset

0,2

Comments

Note: theta_3(x) = Sum_{n=-oo..+oo} x^(n^2) - see A000122.

a(n+1)/a(n) tends to 2.07474... - Vaclav Kotesovec, Feb 05 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 0..361

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

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

(3) theta_3(x) = (x + A(x))/(1 + F(1)), where F(n) = x*(x^(2*n+1) + A(x))/(1 - x*(x^(2*n+1) + A(x)) + F(n+1)), a continued fraction.

(4) 1 - x*theta_3(x) = 1/((1 + x*A(x))*(1 + F(1))), where F(n) = x^(2*n) / (1 - x^(2*n) + x^(2*n+1)*A + (1 + x^(2*n+1)*A)*F(n+1)), a continued fraction.

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 19*x^4 + 40*x^5 + 86*x^6 + 181*x^7 + 383*x^8 + 811*x^9 + 1709*x^10 + 3598*x^11 + 7554*x^12 + ...
By definition, A = A(x) satisfies the sum of products
theta_3(x) = (x + A) - x*(x + A)*(x^3 + A) + x^2*(x + A)*(x^3 + A)*(x^5 + A) - x^3*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A) + x^4*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A)*(x^9 + A) -+ ...
also, A = A(x) satisfies another sum of products
1 - x*theta_3(x) = 1/(1 + x*A) - x^2/((1 + x*A)*(1 + x^3*A)) + x^6/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)) - x^12/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)*(1 + x^7*A)) + x^20/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)*(1 + x^7*A)*(1 + x^9*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
theta_3(x) = (x + A)/(1 + x*(x^3 + A)/(1 - x*(x^3 + A) + x*(x^5 + A)/(1 - x*(x^5 + A) + x*(x^7 + A)/(1 - x*(x^7 + A) + x*(x^9 + A)/(1 - x*(x^9 + A) + x*(x^11 + A)/(1 - ...))))))
where theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + ... + 2*x^(n^2) + ...

Prog

(PARI) {a(n) = my(A=[1], M = sqrtint(n)+1); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=-M, M, x^(n^2) ) - sum(n=0, #A, (-1)^n * x^n * prod(k=0, n, x^(2*k+1) + Ser(A)) ), #A-1) ); H=A; A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A369684, A369682, A369672, A000122 (theta_3).

Sequence in context: A018306 A139784 A247235 * A261663 A199694 A037444

Adjacent sequences: A369680 A369681 A369682 * A369684 A369685 A369686

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2024

Status

approved