cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Ilya Gutkovskiy, Jan 27 2018

Keywords

Examples

			a(5) = 2 because fifth tetrahedral number is 35 and we have [35] and [20, 10, 4, 1].

Links

Crossrefs

Cf. A000292, A030273, A279278, A288126, A298269.

Programs

  • Mathematica
    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}]

Formula

a(n) = [x^A000292(n)] Product_{k>=1} (1 + x^A000292(k)).
a(n) = A279278(A000292(n)).

A337797 Number of partitions of the n-th n-gonal pyramidal number into n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 2, 4, 13, 45, 198, 858, 3728, 16115, 69125, 292940, 1224628, 5052396, 20570806, 82655098, 327881398, 1284663878, 4973614490, 19034194696, 72037124003, 269723590850, 999517370314, 3667158097572, 13325691939021, 47975192145998
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

Examples

			a(3) = 4 because the third tetrahedral (or triangular pyramidal) 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].

Links

Crossrefs

Cf. A006484, A072964, A298269, A337762, A337798, A337799.

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

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

Views

Author

Ilya Gutkovskiy, Nov 08 2020

Keywords

Examples

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

Links

Crossrefs

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

Formula

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

A338778 Number of ordered ways of writing n-th tetrahedral number as a sum of n positive tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 20, 195, 1890, 6286, 94584, 1065120, 12345432, 194450586, 2844976135, 44569913570, 740023110855, 13144353701940, 241663182769494, 4707408836458200, 95865898167054186, 2038122531703155798, 45103282424247100962, 1037559653596650520776, 24776005985596646165127
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 08 2020

Keywords

Examples

			The 5th tetrahedral number is 35 and 35 = 1 + 4 + 10 + 10 + 10 (20 permutations), so a(5) = 20.

Links

Crossrefs

Cf. A000292, A298269, A298672, A298857, A298858, A303170, A338586.

Formula

a(n) = [x^A000292(n)] (Sum_{j>=1} x^A000292(j))^n.

A341773 Number of partitions of 2*n into exactly n nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 4, 2, 0, 4, 3, 0, 5, 4, 1, 5, 4, 1, 5, 5, 1, 6, 6, 2, 6, 6, 2, 6, 7, 3, 7, 9, 4, 8, 9, 4, 8, 10, 5, 9, 12, 6, 10, 12, 7, 10, 13, 8, 12, 15, 10, 13, 16, 11, 13, 17, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Links

Crossrefs

Cf. A000292, A024692, A062748, A068980, A298269, A319799, A338586, A341774, A341775, A341776, A341777, A341778.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1 - x^(Binomial[k + 4, 3] - 1)), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=0} 1 / (1 - x^(binomial(k+4,3)-1)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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