A280718 Expansion of (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

1, 5, 10, 10, 5, 6, 20, 30, 20, 5, 10, 30, 35, 30, 30, 30, 25, 30, 60, 60, 25, 5, 35, 80, 70, 51, 35, 50, 80, 90, 80, 30, 35, 60, 80, 95, 90, 90, 50, 75, 140, 140, 85, 20, 70, 120, 130, 120, 95, 115, 100, 115, 140, 155, 110, 40, 80, 200, 230, 140, 81, 120, 200, 190, 180, 120, 80, 100, 160, 240, 200, 155, 120, 140, 245, 260, 230

Offset: 0

Author

Ilya Gutkovskiy, Feb 10 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 5 pentagonal numbers (A000326).

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

Every number is the sum of at most 5 pentagonal numbers.

Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

Examples

			a(5) = 6 because we have:
[5, 0, 0, 0, 0]
[0, 5, 0, 0, 0]
[0, 0, 5, 0, 0]
[0, 0, 0, 5, 0]
[0, 0, 0, 0, 5]
[1, 1, 1, 1, 1]
		

Links

• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Pentagonal Number
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000326, A003679, A008439, A038671, A280719, A282248.

Programs

Mathematica

nmax = 76; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, 0, nmax}]^5, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(3*k-1)/2))^5.