A218380 -id:A218380

A279281

Expansion of Product_{k>=1} (1 + x^(k*(3*k-2))).

+10
7

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1

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

OFFSET

0,106

COMMENTS

Number of partitions of n into distinct octagonal numbers (A000567).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Octagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

FORMULA

G.f.: Product_{k>=1} (1 + x^(k*(3*k-2))).

EXAMPLE

a(105) = 2 because we have [96, 8, 1] and [65, 40].

MATHEMATICA

nmax = 120; CoefficientList[Series[Product[1 + x^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000567, A024940, A033461, A218380, A279041, A279279, A279280.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 09 2016

STATUS

approved

A279279

Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))).

+10
6

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2

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

OFFSET

0,67

COMMENTS

Number of partitions of n into distinct hexagonal numbers (A000384).

LINKS

Table of n, a(n) for n=0..120.

Eric Weisstein's World of Mathematics, Hexagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

FORMULA

G.f.: Product_{k>=1} (1 + x^(k*(2*k-1))).

EXAMPLE

a(67) = 2 because we have [66, 1] and [45, 15, 6, 1].

MATHEMATICA

nmax = 120; CoefficientList[Series[Product[1 + x^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000384, A024940, A033461, A218380, A278949, A279280, A279281.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 09 2016

STATUS

approved

A279280

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

+10
6

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1

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

OFFSET

0,82

COMMENTS

Number of partitions of n into distinct heptagonal numbers (A000566).

LINKS

Table of n, a(n) for n=0..120.

Eric Weisstein's World of Mathematics, Heptagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

FORMULA

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

EXAMPLE

a(81) = 2 because we have [81] and [55, 18, 7, 1].

MATHEMATICA

nmax = 120; CoefficientList[Series[Product[1 + x^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000566, A024940, A033461, A218380, A279012, A279279, A279281.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 09 2016

STATUS

approved

A281083

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

+10
6

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1

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

OFFSET

0,83

COMMENTS

Number of partitions of n into distinct centered pentagonal numbers (A005891).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from G. C. Greubel)

Eric Weisstein's World of Mathematics, Centered Pentagonal Number

Index entries for sequences related to centered polygonal numbers

Index entries for related partition-counting sequences

FORMULA

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

EXAMPLE

a(82) = 2 because we have [76, 6] and [51, 31].

MATHEMATICA

nmax = 105; CoefficientList[Series[Product[1 + x^(5 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A005891, A218380, A280952, A281081, A281082, A281084.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jan 14 2017

STATUS

approved

A290942

Number of partitions of n into distinct generalized pentagonal numbers (A001318).

+10
5

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 4, 3, 3, 3, 2, 5, 4, 5, 4, 2, 3, 3, 6, 6, 5, 5, 4, 5, 7, 8, 8, 7, 6, 6, 6, 8, 9, 9, 9, 7, 8, 9, 9, 11, 10, 11, 11, 10, 12, 10, 14, 15, 14, 14, 11, 13, 13, 17, 17, 14, 15, 14, 17, 20, 19, 20, 20, 20, 21, 20, 21, 21, 25, 26, 23, 22, 21, 24, 27

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

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..90.

Eric Weisstein's World of Mathematics, Pentagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

FORMULA

G.f.: Product_{k>=1} (1 + x^(k*(3*k-1)/2))*(1 + x^(k*(3*k+1)/2)).

EXAMPLE

a(15) = 3 because we have [15], [12, 2, 1] and [7, 5, 2, 1].

MATHEMATICA

nmax = 90; CoefficientList[Series[Product[(1 + x^(k (3 k - 1)/2)) (1 + x^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000009, A001318, A095699, A218379, A218380, A279221, A280952, A281083.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 14 2017

STATUS

approved

A350196

a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero pentagonal numbers in exactly n ways, or 0 if no such integer exists.

+10
4

1, 35, 92, 127, 144, 214, 237, 215, 249, 250, 319, 315, 354, 355, 366, 390, 391, 432, 431, 425, 475, 448, 478, 460, 482, 483, 510, 495, 537, 531, 525, 545, 570, 560, 594, 566, 581, 582, 606, 601, 595, 618, 603, 630, 602, 625, 652, 666, 657, 641

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..50.

Eric Weisstein's World of Mathematics, Pentagonal Number

CROSSREFS

Cf. A000326, A097563, A218380, A274046, A350197, A350198, A350199.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 19 2021

STATUS

approved

A305355

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

+10
3

1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 0, -1, 2, -1, 0, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0, 0, 1

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

OFFSET

0,36

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Wikipedia, Pentagonal number

MAPLE

seq(coeff(series(mul(1-x^(k*(3*k-1)/2), k=1..n), x, n+1), x, n), n=0..140); # Muniru A Asiru, May 31 2018

CROSSREFS

Product_{k>=1} (1 - x^(k*((m-2)*k-(m-4))/2)): A292518 (m=3), A276516 (m=4), this sequence (m=5).

Cf. A000326, A218379, A218380.

KEYWORD

sign

AUTHOR

Seiichi Manyama, May 31 2018

STATUS

approved

A332007

Number of compositions (ordered partitions) of n into distinct pentagonal numbers.

+10
3

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 2, 7, 2, 0, 0, 6, 26, 6, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 8, 24, 0, 0, 2, 8, 6, 0, 0, 0, 6, 26, 6, 0, 0, 0, 6, 30, 25, 2, 0, 2, 30, 122, 6, 0, 6, 24

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

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..79.

Eric Weisstein's World of Mathematics, Pentagonal Number

Index entries for sequences related to compositions

EXAMPLE

a(18) = 6 because we have [12, 5, 1], [12, 1, 5], [5, 12, 1], [5, 1, 12], [1, 12, 5] and [1, 5, 12].

CROSSREFS

Cf. A000326, A181324, A218379, A218380, A331843, A331844.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 04 2020

STATUS

approved

A296238

Expansion of Product_{k>0} (1 + x^(k*(3*k+1)/2)).

+10
1

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 0

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

OFFSET

0,58

LINKS

Table of n, a(n) for n=0..87.

MATHEMATICA

nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2017 *)

PROG

(PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^(k*(3*k+1)/2))))

CROSSREFS

Cf. A218380, A290942, A296237.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Dec 09 2017

STATUS

approved