A068980 -id:A068980 - 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)).

A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct tetrahedral numbers (A000292).

			a(35) = 2 because we have [35] and [20, 10, 4, 1].

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

Cf. A000292, A007294, A024940, A068980, A350205 (positions of records).

nmax=120; CoefficientList[Series[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

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

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)).

A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

Original entry on oeis.org

1, 1, 2, 4, 11, 29, 94, 304, 1005, 3336, 11398, 38739, 132340, 451086, 1541074, 5242767, 17779666, 60048847, 202124143, 677000711, 2256910444, 7486274436, 24713275977, 81162110629, 265192045408, 862061443031, 2788194736946, 8972104829849, 28726271274133, 91515498561954, 290116750935925

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(3) = 4 because third tetrahedral number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

David A. Corneth, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000292, A037444, A068980, A072964, A298857.

Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 30}]

a(n) = [x^A000292(n)] Product_{k>=1} 1/(1 - x^A000292(k)).

a(n) = A068980(A000292(n)).

A226748 Number of partitions of n into Platonic numbers, cf. A053012.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 14, 14, 20, 20, 26, 27, 37, 37, 46, 47, 62, 63, 77, 80, 101, 103, 125, 130, 160, 164, 194, 203, 245, 253, 296, 311, 368, 381, 440, 463, 540, 562, 642, 677, 780, 814, 922, 973, 1107, 1157, 1302, 1375, 1552, 1626

Offset: 0

Reinhard Zumkeller, Jun 17 2013

nonn

			First Platonic numbers: 1, 4, 6, 8, 10, 12, 19, 20, ...
a(10) = #{10, 8+1+1, 6+4, 6+1+1+1+1, 4+4+1+1, 4+6x1, 10x1} = 7;
a(11) = #{10+1, 8+1+1+1, 6+4+1, 6+5x1, 4+4+1+1+1, 4+7x1, 11x1} = 7;
a(12) = #{12, 10+1+1, 8+4, 8+1+1+1+1, 6+6, 6+4+1+1, 6+6x1, 4+4+4, 4+4+1+1+1+1, 4+8x1, 12x1} = 11;
a(13) = #{12+1, 10+1+1+1, 8+4+1, 8+5x1, 6+6+1, 6+4+1+1+1, 6+7x1, 4+4+4+1, 4+4+5x1, 4+9x1, 13x1} = 11;
a(14) = #{12+1+1, 10+4, 10+1+1+1+1, 8+6, 8+4+1+1, 8+6x1, 6+6+1+1, 6+4+4, 6+4+1+1+1+1, 6+8x1, 4+4+4+1+1, 4+4+6x1, 4+10x1, 14x1} = 14;
a(15) = #{12+1+1+1, 10+4+1, 10+5x1, 8+6+1, 8+4+1+1+1, 8+7x1, 6+6+1+1+1, 6+4+4+1, 6+4+5x1, 6+9x1, 4+4+4+1+1+1, 4+4+7x1, 4+11x1, 15x1} = 14;
a(16) = #{12+4, 12+1+1+1+1, 10+6, 10+4+1+1, 10+6x1, 8+8, 8+6+1+1, 8+4+4, 8+4+1+1+1+1, 8+8x1, 6+6+4, 6+6+1+1+1+1, 6+4+4+1+1, 6+4+6x1, 6+10x1, 4+4+4+4, 4+4+4+1+1+1+1, 4+4+8x1, 4+12x1, 16x1} = 20.

Cf. A003108, A068980, A226749.

a226748 = p a053012_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

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)).

A303170 Number of ordered ways of writing n as a sum of n tetrahedral numbers.

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 141, 309, 757, 2111, 6051, 16721, 44617, 118301, 318501, 871781, 2400741, 6596953, 18067329, 49460555, 135697395, 373271515, 1028451579, 2835353337, 7819016521, 21572619771, 59562583471, 164586609409, 455114644297, 1259191262441, 3485551053561

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

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

Main diagonal of A290429.

Cf. A000292, A068980, A106337, A282582, A303172.

Table[SeriesCoefficient[Sum[x^(k (k + 1) (k + 2)/6), {k, 0, n}]^n, {x, 0, n}], {n, 0, 31}]

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

a(n) = A290429(n,n).

A331919 Number of compositions (ordered partitions) of n into distinct tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 25, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 24, 0, 0, 0, 0, 2, 7, 2, 0, 6, 26, 6, 0, 0, 0, 6, 26, 6, 0, 24, 126, 24, 0, 0, 0, 0, 2, 6, 0, 0, 6, 24, 0, 0, 1, 2, 6, 24, 2, 6, 24

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

			a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A000292, A023361, A068980, A279278, A282582, A298857, A331843, A331984.

N:= 200: # for a(0)..a(N)
G:= mul(1+t*x^(i*(i+1)*(i+2)/6), i=1..floor((6*N)^(1/3))):
F:= proc(n) local R, k, v;
  R:= coeff(G, x, n);
  add(k!*coeff(R, t, k), k=1..degree(R, t))
end proc:
F(0):= 1:
map(F, [$0..N]); # Robert Israel, Feb 03 2020

M = 100;
G = Product[1 + t x^(i(i+1)(i+2)/6), {i, 1, Floor[(6M)^(1/3)]}];
F[n_] := Module[{R, k, v}, R = Coefficient[G, x, n]; Sum[k! Coefficient[R, t, k], {k, 1, Exponent[R, t]}]];
F[0] = 1;
F /@ Range[0, M] (* Jean-François Alcover, Jun 20 2020, after Robert Israel *)

A338586 Number of partitions of the n-th tetrahedral number into exactly n positive tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 5, 5, 20, 35, 75, 154, 336, 730, 1570, 3394, 7339, 16085, 35015, 76269, 164821, 359704, 782004, 1696804, 3668860, 7953962, 17184203, 37093184, 79825297, 171824175, 368838299, 790404448, 1690297309, 3610816466, 7696144659, 16374004711, 34766160358

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 6th tetrahedral number is 56 and 56 = 1 + 1 + 4 + 10 + 20 + 20 = 4 + 4 + 4 + 4 + 20 + 20, so a(6) = 2.

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to partitions

Cf. A000292, A068980, A298269, A298857, A303170, A307643, A338585, A338778.

a(n) = [x^A000292(n) y^n] Product_{j>=1} 1 / (1 - y*x^A000292(j)).

cp's OEIS Frontend

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

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A279278 Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

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

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A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

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A226748 Number of partitions of n into Platonic numbers, cf. A053012.

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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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A303170 Number of ordered ways of writing n as a sum of n tetrahedral numbers.

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A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).

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)).