A327556 Number of colored integer partitions of 2n using all colors of an n-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

1, 1, 15, 319, 10305, 456540, 26189661, 1870454452, 161632399892, 16535827882568, 1968749174314009, 269023182822761584, 41709476698204311667, 7266527579101535573799, 1410853257166617346437587, 303111227353456160724127886, 71611509245127165374518157052

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn

Links

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

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(b(2*n$2, i)*(-1)^(n-i)*C(n, i), i=0..n):
seq(a(n), n=0..17);

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[b[2n, 2n, i] (-1)^(n - i) c[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Formula

a(n) = A327116(2n,n).