A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

1, 1, 3, 19, 121, 1041, 11191, 130663, 1731969, 25778161, 432791371, 7752723771, 151553121193, 3178030999729, 71244609480591, 1716351868658911, 43661944977384961, 1173984102030774753, 33302371396771085779, 991402105480284394531, 30912472614894951462681

Offset

0,3

Comments

Differs from A293840 and from A294253 first at n = 6.

Links

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

Formula

a(n) = A327673(n,n).

Example

a(3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.

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(b(n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..21);

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[b[n, n, i, 0] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

Crossrefs

Main diagonal of A327673.

Cf. A293840, A294253.

Sequence in context: A294252 A294253 A293840 * A318811 A352299 A074572

Adjacent sequences: A327671 A327672 A327673 * A327675 A327676 A327677

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2019

Status

approved