A284641 Expansion of (Sum_{k>=0} x^(k^2*(k+1)^2/4))^12.

1, 12, 66, 220, 495, 792, 924, 792, 495, 232, 198, 672, 1981, 3960, 5544, 5544, 3960, 1980, 726, 792, 2982, 7920, 13860, 16632, 13860, 7920, 2970, 880, 2046, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 727, 4092, 14520, 29700, 38610, 33264, 19404, 7920, 2475, 1584, 6996, 22584, 43560, 55440, 49896

Offset: 0

Author

Ilya Gutkovskiy, May 06 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 12 squares of triangular numbers (A000537).

Every number is the sum of three triangular numbers (Fermat's polygonal number theorem).

Conjecture: a(n) > 0 for all n.

Extended conjecture: every number is the sum of at most 12 squares of triangular numbers (or partial sums of cubes).

Is there a solution, in analogy with Waring's problem (see A002804), for the partial sums of k-th powers?

Links

• Ilya Gutkovskiy, Extended graphical example
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000217, A000537, A014787, A282173, A282288.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^(k^2 (k + 1)^2/4), {k, 0, nmax}]^12, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k^2*(k+1)^2/4))^12.