A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color.

1, 3, 9, 19, 39, 72, 128, 216, 354, 563, 876, 1335, 1998, 2946, 4284, 6154, 8742, 12294, 17129, 23667, 32442, 44151, 59682, 80169, 107054, 142167, 187812, 246895, 323058, 420852, 545958, 705438, 908043, 1164609, 1488504, 1896174, 2407836, 3048255, 3847716

Offset

3,2

Comments

With offset 0 convolution cube of A000009(k+1). - George Beck, Jan 29 2021

Links

Vaclav Kotesovec, Table of n, a(n) for n = 3..10000 (terms 3..5000 from Alois P. Heinz)

Wikipedia, Partition (number theory)

Formula

a(n) ~ exp(Pi*sqrt(n)) / (8 * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^3. - George Beck, Jan 29 2021

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))(3):
seq(a(n), n=3..45);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Crossrefs

Column k=3 of A308680.

Cf. A000009.

Sequence in context: A080010 A135117 A038163 * A146819 A147213 A146441

Adjacent sequences: A327378 A327379 A327380 * A327382 A327383 A327384

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved