A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

1, 1, 7, 94, 2081, 67390, 2969647, 169299808, 12032189630, 1036485156029, 105880393642170, 12604896326749405, 1724189631362670619, 267831346979691504798, 46782781937811822181581, 9111872329195713764645644, 1964607669245374038857479576

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..160

Formula

a(n) = A327117(2n,n).

Example

a(2) = 7: 2ab2ab, 2ab1a1a, 2ab1a1b, 2ab1b1b 1a1a1a1b, 1a1a1b1b, 1a1b1b1b.

Maple

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)*binomial(
      binomial(k, i)+j-1, j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);

Mathematica

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] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Crossrefs

Cf. A327117.

Sequence in context: A243679 A367162 A360474 * A015225 A183521 A342109

Adjacent sequences: A327840 A327841 A327842 * A327844 A327845 A327846

Keyword

nonn

Author

Alois P. Heinz, Sep 27 2019

Status

approved