A178887 Total of n-colorings of parts of all integer partitions of n.

1, 1, 4, 15, 76, 405, 2616, 18613, 151432, 1367649, 13720060, 151005261, 1812987804, 23570657773, 330012270784, 4950230221875, 79204352557936, 1346475340841553, 24236578276301844, 460495032000171373, 9209901462655990180, 193407932383031348241, 4254974546342806648384

Offset

0,3

Comments

An integer partition of n with k parts can have its parts colored in n!/(n-k)! ways. a(n) is the sum of all these possibilities over all integer partitions of n. - Olivier Gérard, May 08 2012

Table A178888 has A000041 entries per row.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..449

Example

A178888 begins
1
2 2
3 6 6
4 12 12 24 24
...
therefore A178887 begins 1 4 15 76 405 ...

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      p!/(p-n)!, b(n, i-1, p)+p*b(n-i, min(i, n-i), p-1))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2019

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, p!/(p - n)!, b[n, i - 1, p] + p b[n - i, Min[i, n - i], p - 1]];
a[n_] := b[n, n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Crossrefs

Row sums of the irregular table A178888.

Sequence in context: A198057 A263004 A002750 * A002467 A332652 A243327

Adjacent sequences: A178884 A178885 A178886 * A178888 A178889 A178890

Keyword

easy,nonn

Author

Alford Arnold, Jun 21 2010

Extensions

More terms and more direct definition by Olivier Gérard, May 08 2012

a(0)=1 prepended by Alois P. Heinz, Jan 21 2019

Status

approved