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.

A259437 a(n) = Sum_{k=0..n} p(k)^n, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 37, 724, 20209, 1905630, 191250531, 57659285287, 20931112851787, 17697850924585423, 17720783665888137843, 44421728434157120665320, 117208746422032553556330253, 679595843556865572365153402674, 4907378683411420479410336076467628
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 27 2015

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[PartitionsP[k]^n,{k,0,n}],{n,0,15}]

Formula

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A259438 a(n) = Sum_{k=0..n} p(k)^(n-k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 3, 5, 10, 25, 78, 301, 1414, 7964, 53408, 426116, 4028890, 44697755, 576491980, 8617031811, 149425700853, 3004591733938, 69763130950599, 1860330686377532, 56746090401472922, 1975156902590115291, 78299783319570477185, 3529323014512112469681
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 27 2015

Keywords

Comments

The position of the maximum value asymptotically approaches k = n/3.

Crossrefs

Programs

  • Mathematica
    Table[Sum[PartitionsP[k]^(n-k),{k,0,n}],{n,0,25}]

Formula

log(a(n)) ~ 2^(3/2)*Pi*n^(3/2)/9 - n*log(16*n^2/3)/3.
G.f.: Sum_{k>=0} x^k/(1 - p(k)*x). - Ilya Gutkovskiy, Oct 09 2018

A265095 a(n) = Sum_{k=0..n} q(k)^k, where q(k) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 2, 3, 11, 27, 270, 4366, 82491, 1762107, 135979835, 10135979835, 753144350523, 130499482241148, 20953464347912316, 6242774737775732860, 2960555481288609431503, 1211886375095917784137679, 719537152598665509899534287, 851154233276178632011679465423
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2015

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[PartitionsQ[k]^k, {k,0,n}], {n,0,20}]

Formula

a(n) ~ exp(n^(3/2)*Pi/sqrt(3) + (Pi/(48*sqrt(3)) - 3*sqrt(3)/(8*Pi))*sqrt(n) - 1/32 - 9/(16*Pi^2)) / (3^(n/4) * 4^n * n^(3*n/4)) ~ q(n)^n.
Showing 1-3 of 3 results.