A327596 Number of colored compositions of 2n using all colors of an n-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

1, 1, 27, 1222, 78819, 7990555, 1075539168, 185948116920, 39826324710186, 10231314625984628, 3097070454570888110, 1088018981038197792790, 436918864329884469153204, 198400793333371519398942287, 100941775818744369615731919906, 57064609834208008799145534143376

Offset: 0

Author

Alois P. Heinz, Sep 18 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A327244.

Programs

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i, 0)*(-1)^(n-i)*C(n, i), i=0..n):
seq(a(n), n=0..17);

Mathematica

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
    b[n-i*j, Min[n-i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, i, 0]*(-1)^(n-i)*c[n, i], {i, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

Formula

a(n) = A327244(2n,n).