A282173 Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.

1, 6, 15, 20, 15, 12, 31, 60, 60, 30, 21, 60, 90, 60, 21, 50, 120, 120, 50, 36, 135, 210, 135, 30, 60, 186, 186, 60, 15, 120, 217, 150, 75, 120, 240, 246, 180, 180, 210, 216, 150, 180, 200, 180, 150, 200, 300, 240, 165, 180, 390, 390, 180, 60, 180, 372, 225, 110, 135, 330, 351, 270, 300, 360, 435, 300, 375, 360, 300, 210

Offset: 0

Author

Ilya Gutkovskiy, Feb 07 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).

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

Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.

Generalized conjecture: every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5). - Ilya Gutkovskiy, Feb 10 2017

Examples

			a(5) = 12 because we have:
[5, 0, 0, 0, 0, 0]
[0, 5, 0, 0, 0, 0]
[0, 0, 5, 0, 0, 0]
[0, 0, 0, 5, 0, 0]
[0, 0, 0, 0, 5, 0]
[0, 0, 0, 0, 0, 5]
[1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 0, 1]
[1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1]
[1, 0, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1]
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Square Pyramidal Number
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000330, A045848.

Programs

Mathematica

nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.