A279278 -id:A279278 - cp's OEIS frontend

A282582 Number of compositions (ordered partitions) of n into tetrahedral (or triangular pyramidal) numbers (A000292).

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 15, 21, 29, 40, 57, 81, 114, 159, 223, 314, 444, 625, 878, 1233, 1736, 2445, 3441, 4838, 6804, 9573, 13473, 18957, 26668, 37514, 52780, 74264, 104488, 147000, 206808, 290961, 409369, 575955, 810314, 1140029, 1603924, 2256603, 3174867, 4466763, 6284339, 8841533, 12439323

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

nonn

			a(8) = 7 because we have [4, 4], [4, 1, 1, 1, 1], [1, 4, 1, 1, 1], [1, 1, 4, 1, 1], [1, 1, 1, 4, 1], [1, 1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to compositions

Cf. A000292, A023361, A068980, A279278, A282172.

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (k + 2)/6), {k, 1, nmax}]), {x, 0, nmax}], x]

Vec(1/(1 - sum(k=1, 50, x^(k*(k + 1)*(k + 2)/6)) + O(x^51))) \\ Indranil Ghosh, Mar 15 2017

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

A350205 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero tetrahedral numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 35, 140, 175, 225, 290, 374, 400, 430, 455, 539, 540, 630, 655, 690, 595, 680, 760, 795, 715, 770, 865, 830, 945, 875, 980, 935, 960, 1080, 990, 970, 1075, 995, 1160, 1165, 1110, 1050, 1170, 1135, 1220, 1190, 1350, 1305, 1285, 1389, 1310, 1320, 1360, 1355, 1340

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A274046, A279278, A350206.

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 1, 0, 0, 1, 2

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

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

			a(46) = 2 because we have [46] and [31, 10, 4, 1].

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, A024940, A279278, A280950, A281082, A281083, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(3 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

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

A298857 Number of partitions of the n-th tetrahedral number into distinct tetrahedral numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 5, 5, 10, 12, 17, 15, 22, 30, 56, 65, 72, 92, 172, 219, 299, 368, 478, 810, 1055, 1508, 1778, 2277, 3815, 5214, 7103, 8615, 11614, 18079, 24428, 33704, 42877, 56639, 85597, 116984, 159179, 199356, 268965, 400612, 545674, 740356, 950897, 1261597, 1842307

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(5) = 2 because fifth tetrahedral number is 35 and 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, A030273, A279278, A288126, A298269.

Table[SeriesCoefficient[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 53}]

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

a(n) = A279278(A000292(n)).

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)).

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 *)

A298247 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, 0, 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, 0, -1, 0, 1, -2, 1, 0, -1, 1, 0, -1, 1, 0, 1

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

sign

The difference between the number of partitions of n into an even number of distinct tetrahedral numbers and the number of partitions of n into an odd number of distinct tetrahedral numbers.

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

Cf. A000292, A068980, A279278, A292518, A298249.

nmax = 104; 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^A000292(k)).

cp's OEIS Frontend

A282582 Number of compositions (ordered partitions) of n into tetrahedral (or triangular pyramidal) numbers (A000292).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350205 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero tetrahedral numbers in exactly n ways, or 0 if no such integer exists.

Views

Author

Keywords

Links

Crossrefs

A281081 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A298857 Number of partitions of the n-th tetrahedral number into distinct tetrahedral numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A298247 Expansion of Product_{k>=1} (1 - x^(k*(k+1)*(k+2)/6)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

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

A298247 Expansion of Product_{k>=1} (1 - x^(k(k+1)(k+2)/6)).