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.

0, 1, 2, 5, 9, 17, 32, 55, 93, 154, 257, 407, 648, 1003, 1546, 2367, 3566, 5323, 7889, 11579, 16854, 24495, 35171, 50345, 71520, 101184, 142118, 198981, 277260, 384457, 530875, 730220, 1000192, 1365105, 1856155, 2514737, 3398397, 4574460, 6141309, 8218229

Offset: 0

Author

Alois P. Heinz, Aug 28 2019

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1650

Crossrefs

Cf. A003056, A321878.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= proc(n) option remember; add(add(binomial(k, i)*(-1)^i*
      b(n$2, k-i), i=0..k)/(k-1)!, k=1..floor((sqrt(1+8*n)-1)/2))
    end:
seq(a(n), n=0..44);

Mathematica

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]]];
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]}];
a /@ Range[0, 44] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..A003056(n)} k * A321878(n,k).