A320931 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

1, 2, 4, 12, 24, 80, 292, 966, 3876, 15554, 61608, 254612, 1065676, 4471672, 19074968, 82043172, 354365492, 1543432514, 6760146292, 29732837780, 131440491584, 583419967664, 2598585783488, 11615321544700, 52079369904384, 234157152231726, 1055628140278948, 4770576024205060

Offset

0,2

Comments

Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n*(n+1)/2.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300 (first 101 terms from Seiichi Manyama)

Formula

a(n) ~ c * d^n / n^(7/4), where d = 4.818071572655... and c = 0.5869031198... - Vaclav Kotesovec, Oct 29 2018

Example

Solutions (a_1, a_2, ... , a_4) to the equation a_1^2 + 2*a_2^2 + ... + 4*a_4^2 = 10.
-------------------------------------------------------------------------------------
( 1,  1,  1,  1), ( 1,  1,  1, -1),
( 1,  1, -1,  1), ( 1,  1, -1, -1),
( 1, -1,  1,  1), ( 1, -1,  1, -1),
( 1, -1, -1,  1), ( 1, -1, -1, -1),
(-1,  1,  1,  1), (-1,  1,  1, -1),
(-1,  1, -1,  1), (-1,  1, -1, -1),
(-1, -1,  1,  1), (-1, -1,  1, -1),
(-1, -1, -1,  1), (-1, -1, -1, -1),
( 2,  1,  0,  1), ( 2,  1,  0, -1),
( 2, -1,  0,  1), ( 2, -1,  0, -1),
(-2,  1,  0,  1), (-2,  1,  0, -1),
(-2, -1,  0,  1), (-2, -1,  0, -1).

Mathematica

nmax = 25; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

Crossrefs

Cf. A000122, A000217, A173519, A320067, A320932.

Sequence in context: A200337 A367173 A367703 * A096421 A066843 A051905

Adjacent sequences: A320928 A320929 A320930 * A320932 A320933 A320934

Keyword

nonn

Author

Seiichi Manyama, Oct 28 2018

Status

approved