login
A320931
a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.
3
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 28 2018
STATUS
approved