A282582 -id:A282582 - cp's OEIS frontend

A341794 Number of ways to write n as an ordered sum of 3 nonzero tetrahedral numbers.

1, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 6, 0, 0, 3, 0, 0, 3, 3, 0, 3, 6, 0, 0, 3, 0, 1, 6, 0, 0, 6, 0, 0, 3, 0, 0, 9, 3, 0, 3, 3, 0, 6, 0, 0, 6, 3, 0, 0, 0, 0, 3, 6, 0, 3, 6, 1, 6, 0, 0, 3, 6, 0, 6, 0, 0, 6, 3, 0, 0, 3, 3, 3, 6, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 6, 3, 0, 9, 0, 0, 12

Offset: 3

Ilya Gutkovskiy, Feb 19 2021

nonn

Cf. A000292, A023533, A023670, A053604, A282582, A341774, A341795, A341796, A341797.

nmax = 95; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^3, {x, 0, nmax}], x] // Drop[#, 3] &

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

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.

A341797 Number of ways to write n as an ordered sum of 6 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 0, 26, 0, 0, 45, 0, 0, 66, 0, 0, 76, 6, 0, 90, 30, 0, 96, 60, 0, 80, 90, 0, 75, 150, 0, 60, 192, 0, 35, 210, 15, 30, 270, 60, 15, 270, 90, 6, 270, 120, 6, 306, 195, 0, 240, 210, 1, 246, 270, 20, 240, 360, 60, 180, 330, 60, 216, 450, 80, 210, 435, 120, 216, 360

Offset: 6

Ilya Gutkovskiy, Feb 19 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 6..1000

Cf. A000292, A023533, A023670, A282582, A340951, A341777, A341794, A341795, A341796, A341797, A341806, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^6 )); // G. C. Greubel, Jul 20 2022

nmax = 77; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(6, x) ).list()
a=A341797_list(100); a[6:81] # G. C. Greubel, Jul 20 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^6.

A341795 Number of ways to write n as an ordered sum of 4 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 6, 0, 0, 8, 0, 0, 13, 0, 0, 12, 0, 0, 10, 4, 0, 12, 12, 0, 6, 12, 0, 4, 16, 0, 4, 24, 0, 0, 16, 0, 1, 24, 6, 0, 24, 12, 0, 16, 6, 0, 28, 12, 0, 12, 12, 0, 12, 12, 0, 16, 30, 4, 12, 12, 4, 16, 24, 0, 16, 24, 4, 24, 6, 0, 12, 24, 12, 12, 18, 12, 13, 36, 0, 0, 24, 12

Offset: 4

Ilya Gutkovskiy, Feb 19 2021

nonn

Cf. A000292, A023533, A023670, A282582, A340949, A341775, A341794, A341796, A341797.

nmax = 85; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^4.

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)).

A341806 Number of ways to write n as an ordered sum of 7 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 0, 42, 0, 0, 77, 0, 0, 126, 0, 0, 168, 7, 0, 211, 42, 0, 252, 105, 0, 252, 182, 0, 245, 315, 0, 231, 469, 0, 175, 574, 21, 140, 735, 105, 105, 854, 210, 56, 875, 315, 42, 987, 525, 21, 952, 693, 7, 882, 840, 42, 924, 1155, 140, 770, 1260, 211, 749, 1470

Offset: 7

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 7..1000

Cf. A000292, A023533, A023670, A282582, A340952, A341778, A341794, A341795, A341796, A341797, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^7 )); // G. C. Greubel, Jul 19 2022

nmax = 72; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341806_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(7, x) ).list()
a=A341806_list(100); a[7:81] # G. C. Greubel, Jul 19 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^7.

A341807 Number of ways to write n as an ordered sum of 8 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 8, 0, 0, 28, 0, 0, 64, 0, 0, 126, 0, 0, 224, 0, 0, 336, 8, 0, 456, 56, 0, 589, 168, 0, 672, 336, 0, 708, 616, 0, 728, 1016, 0, 658, 1400, 28, 560, 1856, 168, 476, 2352, 420, 336, 2632, 728, 238, 2968, 1260, 168, 3192, 1904, 84, 3096, 2464, 112, 3192, 3360, 308, 3024, 4144

Offset: 8

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 8..1000

Cf. A000292, A023533, A023670, A282582, A340953, A341791, A341794, A341795, A341796, A341797, A341806, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^8 )); // G. C. Greubel, Jul 19 2022

nmax = 70; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341807_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(8, x) ).list()
a=A341807_list(100); a[8:81] # G. C. Greubel, Jul 19 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^8.

A341808 Number of ways to write n as an ordered sum of 9 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 9, 0, 0, 36, 0, 0, 93, 0, 0, 198, 0, 0, 378, 0, 0, 624, 9, 0, 918, 72, 0, 1269, 252, 0, 1597, 576, 0, 1836, 1134, 0, 2025, 2025, 0, 2058, 3096, 36, 1926, 4356, 252, 1764, 5877, 756, 1470, 7182, 1512, 1134, 8388, 2772, 882, 9576, 4608, 588, 10035, 6552, 462

Offset: 9

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 9..1000

Cf. A000292, A023533, A023670, A282582, A340954, A341792, A341794, A341795, A341796, A341797, A341806, A341807, A341809.

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^9 )); // G. C. Greubel, Jul 18 2022

nmax = 66; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341808_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(9, x) ).list()
a=A341808_list(100); a[9:71] # G. C. Greubel, Jul 18 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^9.

A341809 Number of ways to write n as an ordered sum of 10 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 10, 0, 0, 45, 0, 0, 130, 0, 0, 300, 0, 0, 612, 0, 0, 1095, 10, 0, 1740, 90, 0, 2565, 360, 0, 3490, 930, 0, 4351, 1980, 0, 5130, 3790, 0, 5680, 6330, 45, 5820, 9540, 360, 5715, 13620, 1260, 5292, 17950, 2880, 4530, 22140, 5670, 3780, 26490, 10170, 2940, 29770, 15840

Offset: 10

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 10..1000

Cf. A000292, A023533, A023670, A282582, A340955, A341793, A341794, A341795, A341796, A341797, A341806, A341807, A341808.

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^10 )); // G. C. Greubel, Jul 18 2022

nmax = 66; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341809_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(10, x) ).list()
a=A341809_list(100); a[10:71]  # G. C. Greubel, Jul 18 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^10.

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).

cp's OEIS Frontend

A341794 Number of ways to write n as an ordered sum of 3 nonzero tetrahedral numbers.

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

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A341797 Number of ways to write n as an ordered sum of 6 nonzero tetrahedral numbers.

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A341795 Number of ways to write n as an ordered sum of 4 nonzero tetrahedral numbers.

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

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A341806 Number of ways to write n as an ordered sum of 7 nonzero tetrahedral numbers.

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A341807 Number of ways to write n as an ordered sum of 8 nonzero tetrahedral numbers.

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A341808 Number of ways to write n as an ordered sum of 9 nonzero tetrahedral numbers.

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