A282172 -id:A282172 - 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)).

A341796 Number of ways to write n as an ordered sum of 5 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 5, 0, 0, 10, 0, 0, 15, 0, 0, 25, 0, 0, 31, 0, 0, 30, 5, 0, 35, 20, 0, 30, 30, 0, 20, 40, 0, 20, 65, 0, 10, 65, 0, 5, 70, 10, 5, 90, 30, 0, 70, 30, 1, 85, 40, 0, 80, 60, 0, 50, 50, 0, 70, 90, 10, 50, 90, 20, 50, 80, 10, 60, 130, 20, 65, 70, 20, 65, 90, 30, 50, 110, 70, 65, 100

Offset: 5

Ilya Gutkovskiy, Feb 19 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1000

Cf. A000292, A023533, A023670, A282172, A282582, A340950, A341776, A341794, A341795, A341797, A341806, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 100);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^5 )); // G. C. Greubel, Jul 20 2022

nmax = 82; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(5, x) ).list()
a=A341796_list(120); a[5:100] # G. C. Greubel, Jul 20 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^5.

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

			Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0,  0,  1,  3,   6,  10,  ...
0,  0,  0,  1,   4,  10,  ...
0,  1,  2,  3,   5,  10,  ...
0,  0,  2,  6,  12,  21,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A104246, A286180, A290054, A290430.
Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).
Main diagonal gives A303170.
Similar to, but different from, A045847.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

A282349 Expansion of (Sum_{k>=0} x^(k(2k^2+1)/3))^7.

Original entry on oeis.org

1, 7, 21, 35, 35, 21, 14, 43, 105, 140, 105, 42, 28, 105, 210, 210, 105, 21, 35, 147, 252, 245, 175, 105, 77, 154, 315, 455, 420, 210, 63, 147, 441, 630, 420, 105, 7, 147, 441, 525, 350, 210, 106, 126, 322, 567, 735, 560, 210, 84, 301, 840, 1050, 630, 210, 49, 315, 875, 980, 630, 245, 35, 245, 707, 1050, 980, 560, 210, 168

Offset: 0

Ilya Gutkovskiy, Feb 12 2017

nonn

Number of ways to write n as an ordered sum of 7 octahedral numbers (A005900).

Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (a(n) > 0 for all n >= 0).

			a(6) = 14 because we have:
[6, 0, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0, 0]
[0, 0, 0, 6, 0, 0, 0]
[0, 0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 1, 0, 1]
[1, 1, 1, 1, 0, 1, 1]
[1, 1, 1, 0, 1, 1, 1]
[1, 1, 0, 1, 1, 1, 1]
[1, 0, 1, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1, 1]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Octahedral Number

Cf. A005900, A282172.

nmax = 68; CoefficientList[Series[Sum[x^(k (2 k^2 + 1)/3), {k, 0, nmax}]^7, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.

A282350 Expansion of (Sum_{k>=0} x^(k(5k^2-5*k+2)/2))^15.

Original entry on oeis.org

1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 470, 315, 1380, 5461, 15015, 30030, 45045, 51480, 45045, 30030, 15015, 5460, 1470, 1575, 8205, 30030, 75075, 135135, 180180, 180180, 135135, 75075, 30030, 8190, 1820, 5565, 30030, 100100, 225225, 360360, 420420, 360360, 225225, 100100, 30030, 5460

Offset: 0

Ilya Gutkovskiy, Feb 12 2017

nonn

Number of ways to write n as an ordered sum of 15 icosahedral numbers (A006564).
Pollock conjectured that every number is the sum of at most 5 tetrahedral numbers and that every number is the sum of at most 7 octahedral numbers.
Conjecture: a(n) > 0 for all n >= 0.
Extended conjecture: every number is the sum of at most 15 icosahedral numbers.

Ilya Gutkovskiy, Extended graphical example

Cf. A006564, A282172, A282349.

nmax = 47; CoefficientList[Series[Sum[x^(k (5 k^2 - 5 k + 2)/2), {k, 0, nmax}]^15, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(5*k^2-5*k+2)/2))^15.

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

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A341796 Number of ways to write n as an ordered sum of 5 nonzero tetrahedral numbers.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

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A282349 Expansion of (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.

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A282350 Expansion of (Sum_{k>=0} x^(k*(5*k^2-5*k+2)/2))^15.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

A282349 Expansion of (Sum_{k>=0} x^(k(2k^2+1)/3))^7.

A282350 Expansion of (Sum_{k>=0} x^(k(5k^2-5*k+2)/2))^15.