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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Wikipedia, Partition (number theory)

Example

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

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 *)

Crossrefs

Column k=2 of A327116.

Sequence in context: A289969 A261442 A078406 * A258348 A262151 A246320

Adjacent sequences: A327595 A327596 A327597 * A327599 A327600 A327601

Keyword

nonn

Author

Alois P. Heinz, Sep 27 2019

Status

approved