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

2,2

Comments

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

Links

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

Wikipedia, Partition (number theory)

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

Example

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

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 *)

Crossrefs

Column k=2 of A308680.

Cf. A000009.

Sequence in context: A011842 A000094 A182377 * A330378 A058578 A261526

Adjacent sequences: A327377 A327378 A327379 * A327381 A327382 A327383

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved