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.

A133018 Partition number of n, raised to power n.

Original entry on oeis.org

1, 1, 4, 27, 625, 16807, 1771561, 170859375, 54875873536, 19683000000000, 17080198121677824, 16985107389382393856, 43439888521963583647921, 113809328043328941786781301, 667840509835890864312744140625, 4816039244598889571670527496421376
Offset: 0

Views

Author

Omar E. Pol, Oct 31 2007

Keywords

Examples

			a(6)=1771561 because the partition number of 6 is 11 and 11^6=1771561.

Crossrefs

Cf. A000312, A058694, A062457, A133032, A259373, A265094. Partition numbers: A000041.

Programs

Formula

a(n) = A000041(n)^n.
a(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). - Vaclav Kotesovec, Jun 23 2015

Extensions

More terms from R. J. Mathar, Jan 13 2008
a(15) from James C. McMahon, Mar 10 2025

A265015 a(n) = A015128(n)^n.

Original entry on oeis.org

1, 2, 16, 512, 38416, 7962624, 4096000000, 4398046511104, 10000000000000000, 48717667557975775744, 451730952053751361306624, 7982572438812891719395180544, 268637376395543538746286686601216, 16132732437821617561429013924830773248
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2015

Keywords

Links

Crossrefs

Cf. A015128, A133018, A156616, A265094.

Programs

  • Mathematica
    nmax = 15; A015128 = Rest[CoefficientList[Series[Product[(1+x^k)/(1-x^k), {k,1,nmax}], {x,0,nmax}], x]]; Flatten[{1, Table[A015128[[n]]^n, {n,1,nmax}]}]

Formula

a(n) ~ exp(Pi*n^(3/2) - sqrt(n)/Pi - 1/(2*Pi^2)) / (8^n * n^n) * (1 - 1/(3*Pi^3*sqrt(n))).

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

Cf. A259436, A265094.

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.

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