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

0,3

Links

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

Example

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

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 *)

Crossrefs

Column k=2 of A309973.

Sequence in context: A056489 A015649 A297189 * A076102 A094282 A124493

Adjacent sequences: A327887 A327888 A327889 * A327891 A327892 A327893

Keyword

nonn

Author

Alois P. Heinz, Sep 29 2019

Status

approved