A282349 Expansion of (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.

1, 7, 21, 35, 35, 21, 14, 43, 105, 140, 105, 42, 28, 105, 210, 210, 105, 21, 35, 147, 252, 245, 175, 105, 77, 154, 315, 455, 420, 210, 63, 147, 441, 630, 420, 105, 7, 147, 441, 525, 350, 210, 106, 126, 322, 567, 735, 560, 210, 84, 301, 840, 1050, 630, 210, 49, 315, 875, 980, 630, 245, 35, 245, 707, 1050, 980, 560, 210, 168

Offset: 0

Author

Ilya Gutkovskiy, Feb 12 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 7 octahedral numbers (A005900).

Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (a(n) > 0 for all n >= 0).

Examples

			a(6) = 14 because we have:
[6, 0, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0, 0]
[0, 0, 0, 6, 0, 0, 0]
[0, 0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 1, 0, 1]
[1, 1, 1, 1, 0, 1, 1]
[1, 1, 1, 0, 1, 1, 1]
[1, 1, 0, 1, 1, 1, 1]
[1, 0, 1, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1, 1]
		

Links

• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Octahedral Number

Crossrefs

Cf. A005900, A282172.

Programs

Mathematica

nmax = 68; CoefficientList[Series[Sum[x^(k (2 k^2 + 1)/3), {k, 0, nmax}]^7, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.