A279220 -id:A279220 - cp's OEIS frontend

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

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)).

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)).

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)).

A322340 Number of compositions (ordered partitions) of n into square pyramidal numbers (A000330).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 114, 153, 205, 274, 365, 487, 651, 871, 1165, 1557, 2080, 2780, 3716, 4967, 6639, 8873, 11860, 15853, 21189, 28320, 37850, 50589, 67618, 90379, 120799, 161456, 215797, 288430, 385512, 515269, 688699

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..7939
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index entries for sequences related to compositions

Cf. A000330, A006456, A279220, A282582, A298246.

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:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(i+1)*(2*i+1)/6), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 49; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A279223 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(5*k-2)/6)).

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, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 16, 16, 16, 16, 17, 17, 18, 18, 20, 20, 20, 20, 21, 21, 22, 22, 24, 24, 25, 25, 26, 26, 27, 27, 29, 29, 31, 31, 32, 32, 33, 33, 35, 35, 37, 37

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero heptagonal pyramidal numbers (A002413).

			a(9) = 2 because we have [8, 1] and [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, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002413, A068980, A279220, A279221, A279222, A279224.

nmax=95; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (5 k - 2)/6)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A298246 Expansion of Product_{k>=1} (1 + x^(k(k+1)(2*k+1)/6)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

nonn

Number of partitions of n into distinct square pyramidal numbers.

			a(91) = 2 because we have [91] and [55, 30, 5, 1].

Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index entries for related partition-counting sequences

Cf. A000330, A033461, A279220, A279278, A289895.

nmax = 104; CoefficientList[Series[Product[1 + x^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000330(k)).

A290792 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)^2(k+2)/12)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 18, 18, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 27, 27, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 45, 45, 47, 47, 49, 49, 52, 52, 54, 54, 57

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

nonn

Number of partitions of n into nonzero 4-dimensional pyramidal numbers (A002415).

			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].

Cf. A000330, A001156, A002415, A279220, A290573.

N:= 100: # for a(0)..a(N)
P:= 1:
for k from 1 do
  e:= k*(k+1)^2*(k+2)/12;
  if e > N then break fi;
  P:= P/(1-x^e);
od:
S:= series(P,x,N+1):
[seq](coeff(S,x,n),n=0..N); # Robert Israel, Aug 28 2019

nmax = 90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)^2 (k + 2)/12)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).

A303172 Number of ordered ways of writing n as a sum of n square pyramidal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 631, 1306, 2806, 6931, 19306, 55070, 150816, 391161, 977501, 2426071, 6141865, 16000186, 42465571, 112950916, 297793651, 776866355, 2015237231, 5233754306, 13668689206, 35908153534, 94633042267, 249398115466, 656105299636, 1723150461561

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Main diagonal of A290430.

Cf. A000330, A066535, A279220, A287617, A303170.

Table[SeriesCoefficient[Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, n}]^n, {x, 0, n}], {n, 0, 32}]

a(n) = [x^n] (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^n.

a(n) = A290430(n,n).

A331984 Number of compositions (ordered partitions) of n into distinct square pyramidal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 24, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 25

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(20) = 6 because we have [14, 5, 1], [14, 1, 5], [5, 14, 1], [5, 1, 14], [1, 14, 5] and [1, 5, 14].

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

Cf. A000330, A279220, A298246, A322340, A331844, A331919.

cp's OEIS Frontend

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

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

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

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A322340 Number of compositions (ordered partitions) of n into square pyramidal numbers (A000330).

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A279223 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(5*k-2)/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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A298246 Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(2*k+1)/6)).

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A290792 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).

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

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

A279223 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(5*k-2)/6)).

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

A298246 Expansion of Product_{k>=1} (1 + x^(k(k+1)(2*k+1)/6)).

A290792 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)^2(k+2)/12)).