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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 (list; graph; refs; listen; history; text; internal format)
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: A129643 A332644 A200337 * A096421 A066843 A051905

Adjacent sequences:  A320928 A320929 A320930 * A320932 A320933 A320934

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 28 2018

STATUS

approved

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Last modified May 18 19:29 EDT 2021. Contains 344002 sequences. (Running on oeis4.)