A282350 Expansion of (Sum_{k>=0} x^(k*(5*k^2-5*k+2)/2))^15.

1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 470, 315, 1380, 5461, 15015, 30030, 45045, 51480, 45045, 30030, 15015, 5460, 1470, 1575, 8205, 30030, 75075, 135135, 180180, 180180, 135135, 75075, 30030, 8190, 1820, 5565, 30030, 100100, 225225, 360360, 420420, 360360, 225225, 100100, 30030, 5460

Offset: 0

Author

Ilya Gutkovskiy, Feb 12 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 15 icosahedral numbers (A006564).

Pollock conjectured that every number is the sum of at most 5 tetrahedral numbers and that every number is the sum of at most 7 octahedral numbers.

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

Extended conjecture: every number is the sum of at most 15 icosahedral numbers.

Links

• Ilya Gutkovskiy, Extended graphical example

Crossrefs

Cf. A006564, A282172, A282349.

Programs

Mathematica

nmax = 47; CoefficientList[Series[Sum[x^(k (5 k^2 - 5 k + 2)/2), {k, 0, nmax}]^15, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(5*k^2-5*k+2)/2))^15.