A218379 -id:A218379 - cp's OEIS frontend

A218380 Number of partitions of n into distinct pentagonal parts.

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

Offset: 0

Antonio Roldán, Oct 27 2012

nonn

			A(98)=3 because 98 = 12 + 35 + 51 = 1 + 5 + 92 = 1 + 5 + 22 + 70 with 1, 5, 22, 70, 92 pentagonal numbers.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..100 from Antonio Roldán)

Cf. A000326, A218379, A290942.

{ for (n=1, 100, m=polcoeff(prod(k=1, truncate(1+sqrt(24*n+1))/6, 1+x^(k*(3*k-1)/2)), n);write("B218380.txt",n, " ",m)) }

a(0) = 1 prepended by Seiichi Manyama, Dec 09 2017

A095699 Number of partitions of n into generalized pentagonal numbers.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 8, 10, 12, 14, 18, 20, 25, 29, 34, 40, 45, 53, 60, 69, 80, 89, 103, 114, 131, 147, 165, 186, 207, 232, 258, 286, 319, 352, 392, 432, 477, 525, 578, 636, 699, 765, 839, 916, 1002, 1093, 1192, 1298, 1413, 1536, 1671, 1810, 1965, 2126, 2304

Offset: 0

Jon Perry, Jul 06 2004

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A001318, A218379, A290942.

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

b(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8; \\ A001318
N=66; x='x+O('x^N);
Vec(1/prod(k=1,N, (1-x^b(k))) )
\\ Joerg Arndt, Oct 13 2014

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

A278949 Expansion of Product_{k>=1} 1/(1 - x^(k(2k-1))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 9, 11, 11, 11, 12, 13, 13, 15, 15, 15, 16, 17, 17, 19, 20, 20, 23, 24, 24, 26, 27, 27, 30, 31, 31, 33, 34, 35, 38, 40, 40, 44, 45, 46, 49, 51, 51, 56, 57, 58, 61, 63, 64, 69, 72, 73, 78, 80, 81, 86, 89, 90, 96, 98, 99, 105, 108, 110, 116, 120, 121, 130

Offset: 0

Ilya Gutkovskiy, Dec 02 2016

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

			a(7) = 2 because we have [6, 1] and [1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000384, A001156, A007294, A037444, A218379.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(2*i-1))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

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

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

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 67, 68, 71, 74, 77, 79, 83, 85, 88, 91, 94, 96, 100

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

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

			a(12) = 3 because we have [6, 6], [6, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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

Cf. A005891, A218379, A279221, A280950, A280951, A280953.

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(((t+1)*5*t+2)/2>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(((i+1)*5*i+2)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax = 88; CoefficientList[Series[Product[1/(1 - x^(5 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

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

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 24, 25, 26, 26, 27, 28, 29, 31, 32, 33, 33, 34, 35, 37, 39, 41, 42, 43, 45, 46, 48, 50, 52, 53, 54, 56, 58, 60, 62, 64, 65, 67, 69, 72, 75, 78

Offset: 0

Ilya Gutkovskiy, Dec 03 2016

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

			a(8) = 2 because we have [7, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000566, A001156, A007294, A037444, A218379, A278949.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(5*t-3)/2>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(5*i-3)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (5 k - 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 10, 10, 11, 11, 11, 12, 12, 12, 14, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 19, 21, 21, 22, 24, 25, 26, 26, 26, 28, 28, 29, 31, 32, 33, 33, 33, 35, 35, 36, 39, 40, 42, 42, 43, 45, 46, 47, 50, 51, 53

Offset: 0

Ilya Gutkovskiy, Dec 04 2016

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

			a(9) = 2 because we have [8, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000567, A001156, A007294, A037444, A218379, A278949, A279012.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(3*t-2)>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(3*i-2))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

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

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

A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

nonn

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

Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

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

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]

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

A305355 Expansion of Product_{k>=1} (1 - x^(k(3k-1)/2)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 31 2018

sign

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

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.

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

A332007 Number of compositions (ordered partitions) of n into distinct pentagonal numbers.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			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].

Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for sequences related to compositions

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

A296237 Expansion of Product_{k>0} 1/(1 - x^(k(3k+1)/2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 5, 4, 6, 4, 6, 5, 6, 6, 7, 7, 7, 7, 9, 8, 10, 9, 11, 10, 11, 12, 12, 13, 13, 14, 15, 14, 17, 16, 19, 18, 20, 20, 21, 22, 23, 24, 25, 25, 28, 27, 30, 29, 33, 32, 35, 35, 37, 38, 39

Offset: 0

Seiichi Manyama, Dec 09 2017

nonn

Integer partitions into second or "negative" pentagonal numbers (A005449) .

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

Cf. A005449, A095699, A218379, A296238.

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

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

cp's OEIS Frontend

A218380 Number of partitions of n into distinct pentagonal parts.

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A095699 Number of partitions of n into generalized pentagonal numbers.

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A278949 Expansion of Product_{k>=1} 1/(1 - x^(k*(2*k-1))).

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A280952 Expansion of Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

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A279012 Expansion of Product_{k>=1} 1/(1 - x^(k*(5*k-3)/2)).

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A279041 Expansion of Product_{k>=1} 1/(1 - x^(k*(3*k-2))).

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A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

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A305355 Expansion of Product_{k>=1} (1 - x^(k*(3*k-1)/2)).

A278949 Expansion of Product_{k>=1} 1/(1 - x^(k(2k-1))).

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).

A305355 Expansion of Product_{k>=1} (1 - x^(k(3k-1)/2)).

A296237 Expansion of Product_{k>0} 1/(1 - x^(k(3k+1)/2)).