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

0,3

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A003056, A321878.

Sequence in context: A285458 A000569 A292168 * A182992 A115851 A163734

Adjacent sequences: A322301 A322302 A322303 * A322305 A322306 A322307

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2019

Status

approved