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

%I A178887 #17 Jul 09 2023 18:32:27
%S A178887 1,1,4,15,76,405,2616,18613,151432,1367649,13720060,151005261,
%T A178887 1812987804,23570657773,330012270784,4950230221875,79204352557936,
%U A178887 1346475340841553,24236578276301844,460495032000171373,9209901462655990180,193407932383031348241,4254974546342806648384
%N A178887 Total of n-colorings of parts of all integer partitions of n.
%C A178887 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 A178887 Table A178888 has A000041 entries per row.
%H A178887 Alois P. Heinz, <a href="/A178887/b178887.txt">Table of n, a(n) for n = 0..449</a>
%e A178887 A178888 begins
%e A178887 1
%e A178887 2 2
%e A178887 3 6 6
%e A178887 4 12 12 24 24
%e A178887 ...
%e A178887 therefore A178887 begins 1 4 15 76 405 ...
%p A178887 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A178887       p!/(p-n)!, b(n, i-1, p)+p*b(n-i, min(i, n-i), p-1))
%p A178887     end:
%p A178887 a:= n-> b(n$3):
%p A178887 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 21 2019
%t A178887 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 A178887 a[n_] := b[n, n, n];
%t A178887 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y A178887 Row sums of the irregular table A178888.
%K A178887 easy,nonn
%O A178887 0,3
%A A178887 _Alford Arnold_, Jun 21 2010
%E A178887 More terms and more direct definition by _Olivier Gérard_, May 08 2012
%E A178887 a(0)=1 prepended by _Alois P. Heinz_, Jan 21 2019