A327598 Number of colored integer partitions of n using all colors of a 2-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

0, 0, 2, 6, 15, 32, 65, 124, 230, 414, 729, 1258, 2141, 3586, 5935, 9716, 15738, 25258, 40196, 63452, 99426, 154732, 239219, 367592, 561602, 853300, 1289777, 1939920, 2904003, 4327672, 6421572, 9489260, 13967003, 20479638, 29919253, 43556102, 63193528

Offset: 0

Author

Alois P. Heinz, Sep 27 2019

Keywords

• nonn

Examples

			a(2) = 2: 2ab, 1a1b.
a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000
• Wikipedia, Partition (number theory)

Crossrefs

Column k=2 of A327116.

Programs

Maple

C:= binomial:
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)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k))(2):
seq(a(n), n=0..37);

Mathematica

c = Binomial;
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] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)