A327380 Number of colored integer partitions of n such that two colors are used and parts differ by size or by color.

1, 2, 5, 8, 14, 22, 34, 50, 73, 104, 146, 202, 275, 372, 498, 660, 868, 1134, 1470, 1896, 2430, 3098, 3931, 4964, 6240, 7814, 9746, 12110, 14997, 18510, 22772, 27934, 34166, 41672, 50698, 61520, 74470, 89940, 108378, 130312, 156364, 187244, 223785, 266962

Offset: 2

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Comments

With offset 0 convolution square of A000009(k+1). - George Beck, Jan 28 2021

Examples

			a(4) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
a(5) = 8: 2a2b1a, 2a2b1b, 3a1a1b, 3b1a1b, 3a2b, 3b2a, 4a1b, 4b1a.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 2..10000 (terms 2..5000 from Alois P. Heinz)
• Wikipedia, Partition (number theory)

Crossrefs

Column k=2 of A308680.

Cf. A000009.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..45);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 45] (* Jean-François Alcover, May 06 2020, after Maple *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{j>=1} (1 + x^j))^2. - Alois P. Heinz, Jan 29 2021