A327589 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 have i colors in (weakly) increasing order.

1, 1, 39, 2272, 284319, 56455146, 16786728000, 6935657012558, 3810209706509775, 2684955985258788274, 2361563245536690165774, 2535933313556764621139740, 3265213763332455703665035736, 4965602758384602312429712415116, 8805913731971382862369182854094726

Offset: 0

Author

Alois P. Heinz, Sep 17 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..190 (terms 0..120 from Alois P. Heinz)

Crossrefs

Cf. A327245.

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)*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..15);

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]*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, 15}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

Formula

a(n) = A327245(2n,n).

a(n) ~ c * d^n * n^(2*n), where d = 1.31520176578651896001... and c = 1.569966657460754514... - Vaclav Kotesovec, Sep 19 2019