A280952 -id:A280952 - cp's OEIS frontend

A280950 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)/2+1)).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 11, 11, 12, 13, 15, 15, 16, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 37, 40, 43, 45, 48, 51, 54, 56, 60, 63, 67, 70, 76, 80, 84, 87, 93, 97, 102, 106, 113, 118, 125, 130, 138, 143, 151, 157, 166, 172, 181, 189, 200, 207, 217, 225, 237, 245, 257, 267, 280

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered triangular numbers (A005448).

			a(8) = 3 because we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Robert Israel, Table of n, a(n) for n = 0..10000
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 Triangular Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005448, A007294, A068980, A280951, A280952, A280953.

N:= 100:
kmax:= floor((sqrt(24*N-15)-3)/6):
S:= series(mul(1/(1-x^(3*k*(k+1)/2+1)),k=0..kmax),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 25 2017

nmax = 78; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

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

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 10, 11, 11, 12, 12, 14, 15, 15, 16, 16, 18, 19, 19, 21, 22, 24, 26, 26, 28, 29, 31, 33, 33, 35, 36, 39, 42, 43, 45, 47, 50, 53, 54, 56, 58, 61, 65, 66, 69, 72, 76, 81, 83, 86, 89, 93, 98, 100, 103, 107, 112, 118, 121, 125, 130, 136, 142, 146

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered square numbers (A001844).

			a(10) = 3 because we have [5, 5], [5, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

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 Square Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A001156, A001844, A279220, A280950, A280952, A280953.

nmax = 82; CoefficientList[Series[Product[1/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

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

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 22, 23, 24, 25, 25, 27, 27, 29, 30, 31, 32, 32, 34, 34, 36, 37, 38, 39, 40, 43, 44, 46, 47, 48, 50, 51, 54, 55, 57, 58, 59

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered hexagonal numbers (A003215).

			a(14) = 3 because we have [7, 7], [7, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 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, Hex Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A003215, A278949, A279222, A280950, A280951, A280952.

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

nmax = 86; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

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

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

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

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

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

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

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)).

A322801 Number of compositions (ordered partitions) of n into centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 28, 36, 46, 59, 76, 98, 128, 167, 217, 281, 363, 468, 605, 784, 1017, 1320, 1712, 2217, 2869, 3713, 4807, 6227, 8070, 10458, 13549, 17549, 22726, 29430, 38117, 49375, 63962, 82859, 107333, 139026, 180071

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

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

Cf. A005891, A181324, A280863, A280952, A281083, A282502, A282504, A322802, A322803.

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:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-((i+1)*5*i+2)/2), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(5 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

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

cp's OEIS Frontend

A280950 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+1)).

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A280951 Expansion of Product_{k>=0} 1/(1 - x^(2*k*(k+1)+1)).

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A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

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A281083 Expansion of Product_{k>=0} (1 + x^(5*k*(k+1)/2+1)).

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Mathematica

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

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A322801 Number of compositions (ordered partitions) of n into centered pentagonal numbers (A005891).

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A280950 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)/2+1)).

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).