%I #17 Jul 09 2023 18:32:27

%S 1,1,4,15,76,405,2616,18613,151432,1367649,13720060,151005261,

%T 1812987804,23570657773,330012270784,4950230221875,79204352557936,

%U 1346475340841553,24236578276301844,460495032000171373,9209901462655990180,193407932383031348241,4254974546342806648384

%N Total of n-colorings of parts of all integer partitions of n.

%C 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

%C Table A178888 has A000041 entries per row.

%H Alois P. Heinz, <a href="/A178887/b178887.txt">Table of n, a(n) for n = 0..449</a>

%e A178888 begins

%e 1

%e 2 2

%e 3 6 6

%e 4 12 12 24 24

%e ...

%e therefore A178887 begins 1 4 15 76 405 ...

%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,

%p p!/(p-n)!, b(n, i-1, p)+p*b(n-i, min(i, n-i), p-1))

%p end:

%p a:= n-> b(n$3):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 21 2019

%t 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]];

%t a[n_] := b[n, n, n];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)

%Y Row sums of the irregular table A178888.

%K easy,nonn

%O 0,3

%A _Alford Arnold_, Jun 21 2010

%E More terms and more direct definition by _Olivier Gérard_, May 08 2012

%E a(0)=1 prepended by _Alois P. Heinz_, Jan 21 2019