A299032 -id:A299032 - cp's OEIS frontend

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

1, 1, 0, 3, 18, 60, 252, 1576, 10494, 64152, 458400, 3407019, 27713928, 225193982, 1980444648, 17626414158, 165796077562, 1593587604441, 15985672426992, 163422639872978, 1729188245991060, 18743981599820280, 208963405365941380, 2378065667103672024, 27742569814633730608

Offset: 0

Ilya Gutkovskiy, Feb 01 2018

nonn

			a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4].

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298329, A298858, A298938, A299032.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n.

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 13 2018

nonn

Index entries for sequences related to sums of squares

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

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}]]

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.

cp's OEIS Frontend

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

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