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A322304 Total number of colors in all partitions of n into colored blocks of equal parts, such that all colors from a given set are used and the colors are introduced in increasing order.

%I A322304 #32 Dec 14 2020 07:50:53
%S A322304 0,1,2,5,9,17,32,55,93,154,257,407,648,1003,1546,2367,3566,5323,7889,
%T A322304 11579,16854,24495,35171,50345,71520,101184,142118,198981,277260,
%U A322304 384457,530875,730220,1000192,1365105,1856155,2514737,3398397,4574460,6141309,8218229
%N A322304 Total number of colors in all partitions of n into colored blocks of equal parts, such that all colors from a given set are used and the colors are introduced in increasing order.
%H A322304 Alois P. Heinz, <a href="/A322304/b322304.txt">Table of n, a(n) for n = 0..1650</a>
%F A322304 a(n) = Sum_{k=1..A003056(n)} k * A321878(n,k).
%e A322304 a(4) = 9. The colored partitions are: 1111a, 2a11a, 22a, 3a1a, 4a, 2a11b, 3a1b.  The total number of colors used is 1+1+1+1+1+2+2 = 9.
%p A322304 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A322304      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A322304     end:
%p A322304 a:= proc(n) option remember; add(add(binomial(k, i)*(-1)^i*
%p A322304       b(n$2, k-i), i=0..k)/(k-1)!, k=1..floor((sqrt(1+8*n)-1)/2))
%p A322304     end:
%p A322304 seq(a(n), n=0..44);
%t A322304 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
%t A322304 a[n_] := Sum[Sum[Binomial[k, i] (-1)^i b[n, n, k - i], {i, 0, k}]/(k - 1)!, {k, 1, Floor[(Sqrt[1 + 8n] - 1)/2]}];
%t A322304 a /@ Range[0, 44] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A322304 Cf. A003056, A321878.
%K A322304 nonn
%O A322304 0,3
%A A322304 _Alois P. Heinz_, Aug 28 2019