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.

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

Views

Author

Ilya Gutkovskiy, Jan 27 2018

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

a(n) = [x^A000292(n)] Product_{k>=1} 1/(1 - x^A000292(k)).
a(n) = A068980(A000292(n)).

A337798 Number of partitions of the n-th n-gonal pyramidal number into distinct n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 5, 4, 5, 7, 11, 9, 4, 12, 12, 24, 23, 42, 59, 64, 58, 124, 206, 212, 168, 377, 539, 703, 873, 1122, 1505, 1943, 2724, 4100, 4513, 6090, 7138, 12079, 16584, 20240, 27162, 35874, 52622, 69817, 88059, 115628, 152756, 219538, 240200, 358733, 480674
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

Examples

			a(9) = 2 because the ninth 9-gonal pyramidal number is 885 and we have [885] and [420, 266, 155, 34, 10].

Links

Crossrefs

Cf. A006484, A288126, A298857, A337763, A337797, A337799.

Programs

  • Maple
    p:= (n,k) ->  k * (k + 1) * (k * (n - 2) - n + 5) / 6:
f:= proc(n) local k, P;
  P:= mul(1+x^p(n,k),k=1..n);
  coeff(P,x,p(n,n));
end proc:
map(f, [$0..80]); # Robert Israel, Sep 23 2020
  • PARI
    default(parisizemax, 2^31);
p(n,k) = k*(k + 1)*(k*(n-2) - n + 5)/6;
a(n) = my(f=1+x*O(x^p(n,n))); for(k=1, n, f*=1+x^p(n,k)); polcoeff(f, p(n,n)); \\ Jinyuan Wang, Dec 21 2021

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} (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.

Extensions

More terms from Robert Israel, Sep 23 2020

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

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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

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.
Showing 1-5 of 5 results.

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