A279223 -id:A279223 - cp's OEIS frontend

A279220 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k+1)/6)).

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 9, 9, 10, 11, 13, 13, 13, 14, 15, 17, 17, 17, 18, 19, 21, 21, 22, 23, 25, 27, 27, 28, 29, 31, 33, 33, 34, 35, 37, 40, 41, 42, 44, 46, 50, 51, 52, 54, 56, 60, 61, 62, 64, 67, 72, 73, 75, 77, 81, 86, 87, 89, 91, 95, 100, 101, 103, 106, 111, 117, 119, 121, 125, 130, 137

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero square pyramidal numbers (A000330).

			a(6) = 2 because we have [5, 1] and [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, Square Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000330, A068980, A279221, A279222, A279223, A279224.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(t+1)*(2*t+1)/6>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*(i+1)*(2*i+1)/6)))
    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 (k + 1) (2 k + 1)/6)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).

A279221 Expansion of Product_{k>=1} 1/(1 - x^(k^2*(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 12, 12, 12, 12, 13, 13, 16, 16, 16, 16, 17, 17, 20, 20, 20, 20, 21, 21, 25, 25, 25, 25, 27, 27, 31, 31, 31, 31, 33, 33, 37, 37, 37, 37, 39, 39, 44, 44, 44, 45, 48, 48, 53, 53, 54, 55, 58, 58, 63, 63, 64, 65, 68, 68, 74

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero pentagonal pyramidal numbers (A002411).

			a(7) = 2 because we have [6, 1] and [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, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002411, A068980, A279220, A279222, A279223, A279224.

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

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

A279222 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(4*k-1)/6)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 13, 15, 16, 16, 16, 16, 16, 17, 19, 20, 20, 20, 20, 20, 21, 23, 24, 25, 25, 25, 25, 26, 28, 30, 31, 31, 31, 31, 32, 34, 36, 37, 37, 37, 37, 38, 40, 42, 43, 44, 44, 44

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero hexagonal pyramidal numbers (A002412).

			a(8) = 2 because we have [7, 1] and [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, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002412, A068980, A279220, A279221, A279223, A279224.

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (4 k - 1)/6)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(4*k-1)/6)).

A279224 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k-1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 24, 24, 26, 26, 26, 27, 27, 27, 28, 29, 29, 31, 32

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero octagonal pyramidal numbers (A002414).

			a(10) = 2 because we have [9, 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]
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002414, A068980, A279220, A279221, A279222, A279223.

nmax=100; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (2 k - 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A322855 Number of compositions (ordered partitions) of n into heptagonal pyramidal numbers (A002413).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 97, 121, 151, 188, 233, 287, 352, 431, 530, 654, 809, 1002, 1241, 1535, 1895, 2335, 2876, 3544, 4371, 5396, 6666, 8237, 10176, 12564, 15504, 19126, 23594, 29111, 35928

Offset: 0

Ilya Gutkovskiy, Dec 29 2018

nonn

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

Cf. A002413, A279223, A282582, A322340, A322799, A322803, A322853, A322854, A322856.

nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (5 k - 2)/6), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)*(5*k-2)/6)).

cp's OEIS Frontend

A279220 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).

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A279221 Expansion of Product_{k>=1} 1/(1 - x^(k^2*(k+1)/2)).

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A279222 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(4*k-1)/6)).

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A279224 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k-1)/2)).

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A322855 Number of compositions (ordered partitions) of n into heptagonal pyramidal numbers (A002413).

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A279220 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k+1)/6)).

A279222 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(4*k-1)/6)).

A279224 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k-1)/2)).