A328158 Number of colored integer partitions of 2n using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.

1, 2, 22, 428, 11595, 416010, 18283208, 945843148, 58252818659, 4087684096527, 317934667075551, 28164509102578546, 2781331187964705790, 294700331738309167806, 33811410232219114946609, 4297801013746798965557794, 593679426174377865941838598

Offset: 0

Author

Alois P. Heinz, Oct 05 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..175
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A326500.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..18);

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[k + i j - 1, i j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, n-i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, May 08 2020, after Maple *)

Formula

a(n) = A326500(2n,n).