A327557 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

0, 1, 5, 29, 173, 1129, 7933, 59757, 480389, 4102233, 37059485, 352891285, 3530465753, 37001007337, 405191214949, 4625525704837, 54929552638957, 677283511701937, 8655757492783861, 114479050583748677, 1564613481125976373, 22068492671782019793

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A327116.

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-1, i), 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..23);

Mathematica

c = Binomial;
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] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A327116(n,k).