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 1-3 of 3 results.

A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 43, 85, 164, 308, 573, 1079, 2081, 4097, 8129, 16049, 31315, 60402, 115806, 222416, 430791, 843987, 1670054, 3322167, 6606936, 13078586, 25714238, 50230292, 97708338, 189921842, 370216757, 725680489, 1431888173, 2842060970, 5662371069
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 01 2017

Keywords

Comments

Number of compositions of n into a triangular number of parts.

Examples

			a(5) = 7 because we have:
  [1]  [5]
  [2]  [3, 1, 1]
  [3]  [1, 3, 1]
  [4]  [1, 1, 3]
  [5]  [2, 2, 1]
  [6]  [2, 1, 2]
  [7]  [1, 2, 2]

Links

Crossrefs

Cf. A000217, A052467, A103198, A280351.

Programs

  • Mathematica
    nmax = 36; Rest[CoefficientList[Series[Sum[(x/(1 - x))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 40; Rest[CoefficientList[Series[-1 + EllipticTheta[2, 0, Sqrt[x/(1-x)]]/(2*(x/(1-x))^(1/8)), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)
  • PARI
    a(n) = sum(k=1, (sqrtint(8*n+1)-1)\2, binomial(n-1, k*(k+1)/2-1)) \\ Andrew Howroyd, Jan 14 2023

Formula

G.f.: Sum_{k>=1} (x/(1-x))^(k*(k+1)/2).
a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} binomial(n-1, k*(k+1)/2-1). - Jerzy R Borysowicz, Dec 26 2022
Conjecture: a(n+1)/a(n) ~ 2. - Jerzy R Borysowicz, Jan 14 2023
Conjecture: abs(b(n)-1) < 0.015, where b(n) = a(n)*sqrt(n)/2^(n-1), for n > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023

A334626 G.f.: Sum_{k>=0} x^(k^3) / Product_{j=1..k^3} (1 - x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 30, 41, 53, 71, 90, 117, 147, 187, 231, 289, 354, 436, 528, 642, 770, 927, 1102, 1313, 1550, 1832, 2147, 2519, 2935, 3421, 3964, 4594, 5298, 6110, 7016, 8055, 9216, 10542, 12021, 13706, 15588, 17724, 20111
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2020

Keywords

Comments

Number of partitions of n such that the number of parts is a cube.
Also number of partitions of n such that the largest part is a cube.

Examples

			a(10) = 3 because we have [10], [3, 1, 1, 1, 1, 1, 1, 1] and [2, 2, 1, 1, 1, 1, 1, 1] (see the first comment) or [8, 2], [8, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (see the second comment).

Links

Crossrefs

Cf. A000578, A003108, A089333, A280351, A339235.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[Sum[x^(k^3)/Product[1 - x^j, {j, 1, k^3}], {k, 0, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]

A369406 a(n) = Sum_{k=0..n} binomial(n,k^3).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 19, 56, 177, 508, 1301, 3018, 6451, 12887, 24328, 43777, 75602, 125991, 203512, 319793, 490338, 735496, 1081601, 1562302, 2220104, 3108162, 4292581, 5857016, 7920222, 10719709, 14991758, 23535855, 47071676, 124403657, 386938194, 1252225819
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2024

Keywords

Comments

a(n) equals the number of subsets of [n] whose cardinalities are cube.
Binomial transform of the characteristic function of cubes A010057.
Partial sums of A280351.

Crossrefs

Cf. A010057, A003099, A280351.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k^3], {k, 0, n^(1/3)}], {n, 0, 38}]
nmax = 38; CoefficientList[Series[(1/(1 - x)) Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^3).
Showing 1-3 of 3 results.

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