A282248 -id:A282248

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A280718

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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).

LINKS

Table of n, a(n) for n=0..76.

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Pentagonal Number

Index to sequences related to polygonal numbers

FORMULA

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

EXAMPLE

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]

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 10 2017

STATUS

approved

A280719

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

+10
2

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).

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

Every number is the sum of at most 6 hexagonal numbers.

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

LINKS

Table of n, a(n) for n=0..70.

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Hexagonal Number

Index to sequences related to polygonal numbers

FORMULA

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

EXAMPLE

a(6) = 7 because we have:

[6, 0, 0, 0, 0, 0]

[0, 6, 0, 0, 0, 0]

[0, 0, 6, 0, 0, 0]

[0, 0, 0, 6, 0, 0]

[0, 0, 0, 0, 6, 0]

[0, 0, 0, 0, 0, 6]

[1, 1, 1, 1, 1, 1]