A319221 Number of ordered ways of writing n-th triangular number as a sum of n squares.

1, 2, 0, 24, 144, 960, 4608, 74048, 859952, 9568800, 109975680, 1647979872, 23917274304, 358378620704, 5528847787008, 94307761212304, 1632598198916544, 29205907283227776, 538335591996965760, 10388234139989630128, 205386383159397554688, 4173254005731822569088

Offset: 0

Author

Ilya Gutkovskiy, Sep 13 2018

Keywords

• nonn

Links

• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000217, A000290, A066535, A196010, A232173, A278340, A298858, A299031, A299032.

Programs

Mathematica

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, n (n + 1)/2}], {n, 0, 21}]
Join[{1}, Table[SquaresR[n, n (n + 1)/2], {n, 21}]]

Formula

a(n) = [x^(n*(n+1)/2)] theta_3(x)^n, where theta_3() is the Jacobi theta function.

a(n) = [x^(n*(n+1)/2)] (Sum_{k=-infinity..infinity} x^(k^2))^n.