A327890 Number of colored integer partitions of n using all colors of a 2-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.

0, 0, 3, 6, 21, 42, 90, 176, 348, 640, 1203, 2152, 3848, 6692, 11701, 19968, 33966, 56952, 95300, 157326, 258736, 421240, 683804, 1099830, 1762867, 2805154, 4446826, 7005486, 10999634, 17172894, 26716627, 41362952, 63837722, 98079482, 150216194, 229155682

Offset: 0

Author

Alois P. Heinz, Sep 29 2019

Keywords

• nonn

Examples

			a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..4000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Column k=2 of 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-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..44);

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_] := With[{k = 2}, Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}]];
a /@ Range[0, 44] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)