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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..175

Wikipedia, Partition (number theory)

Formula

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

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 *)

Crossrefs

Cf. A326500.

Sequence in context: A264839 A241347 A163436 * A266522 A360304 A354943

Adjacent sequences: A328155 A328156 A328157 * A328159 A328160 A328161

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2019

Status

approved