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

Author

Alford Arnold, Jun 21 2010

Keywords

• easy
• nonn

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.

Examples

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

Links

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

Crossrefs

Row sums of the irregular table A178888.

Programs

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 *)

Extensions

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

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