A327382 Number of colored integer partitions of n such that four colors are used and parts differ by size or by color.

1, 4, 14, 36, 85, 180, 360, 680, 1234, 2160, 3674, 6092, 9882, 15724, 24594, 37884, 57553, 86344, 128060, 187948, 273178, 393516, 562158, 796860, 1121375, 1567336, 2176664, 3004692, 4124130, 5630160, 7646916, 10335696, 13905376, 18625564, 24843142, 33003072

Offset: 4

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 4..10000 (terms 4..5000 from Alois P. Heinz)
• Wikipedia, Partition (number theory)

Crossrefs

Column k=4 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))(4):
seq(a(n), n=4..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 = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[4, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Formula

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^4. - George Beck, Jan 29 2021