A327681 Number of colored integer partitions of 2n using all colors of an n-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

1, 1, 21, 619, 32621, 2619031, 298688151, 45747815408, 9130881915237, 2302153903685914, 716914926484850891, 270654298469985496639, 121905995767297357401683, 64616493201145984241278851, 39838866068219563302546530228, 28277347692301453998991014108124

Offset: 0

Author

Alois P. Heinz, Sep 21 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..100 from Alois P. Heinz)

Crossrefs

Cf. A309973.

Programs

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-1, i), j)*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 - 1, i], j]*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 *)

Formula

a(n) = A309973(2n,n).