A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order.

0, 1, 5, 24, 129, 752, 4796, 33117, 246336, 1961233, 16626100, 149376533, 1416602126, 14130107135, 147781380186, 1616110614723, 18434515499407, 218849548323400, 2698686223271769, 34504328470389166, 456669361749612835, 6247290917385938422, 88216873775207493056

Offset: 0

Author

Alois P. Heinz, Sep 13 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..230
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A327117.

Programs

Maple

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k, i]+j-1, j], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *)

Formula

a(n) = Sum_{k=1..n} k * A327117(n,k).