A327675 Number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order).

1, 1, 4, 40, 355, 4425, 69521, 1162951, 22259414, 478096938, 11614132907, 299700810545, 8456607358157, 255883964141333, 8275199908539114, 287869753459458468, 10564476589147507523, 409845503129745719513, 16777378294629533764699, 720626728499888159724831

Offset: 0

Author

Alois P. Heinz, Sep 21 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Row sums of A327673.

Programs

Maple

b:= proc(n, i, k, p) option remember;
     `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_, p_] := b[n, i, k, p] =
     If[n == 0, p!, If[i < 1, 0, Sum[ Binomial[k^i, j]*
     b[n - i*j, Min[n - i*j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i, 0]*(-1)^(k - i)*
     Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)