A282172 Expansion of (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5.

1, 5, 10, 10, 10, 21, 30, 20, 15, 30, 35, 30, 40, 40, 35, 60, 65, 25, 30, 60, 46, 50, 80, 50, 55, 120, 95, 20, 60, 90, 60, 80, 100, 40, 80, 145, 85, 30, 90, 85, 105, 155, 100, 40, 155, 170, 90, 80, 100, 90, 171, 145, 40, 60, 140, 110, 125, 130, 80, 140, 250, 170, 70, 110, 140, 160, 190, 140, 90, 180, 220, 170, 95, 70, 110, 215

Offset: 0

Author

Ilya Gutkovskiy, Feb 07 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 5 tetrahedral (or triangular pyramidal) numbers (A000292).

a(n) > 0 for all n ("Pollock's Conjecture").

Examples

			a(4) = 10 because we have:
[4, 0, 0, 0, 0]
[0, 4, 0, 0, 0]
[0, 0, 4, 0, 0]
[0, 0, 0, 4, 0]
[0, 0, 0, 0, 4]
[1, 1, 1, 1, 0]
[1, 1, 1, 0, 1]
[1, 1, 0, 1, 1]
[1, 0, 1, 1, 1]
[0, 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, Pollock's Conjecture
• Eric Weisstein's World of Mathematics, Tetrahedral Number
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000292, A000797, A008439, A104246.

Programs

Mathematica

nmax = 75; CoefficientList[Series[(Sum[x^(k (k + 1) (k + 2)/6), {k, 0, nmax}])^5, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5.