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

4,2

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)

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

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 *)

Crossrefs

Column k=4 of A308680.

Cf. A000009.

Sequence in context: A061989 A079908 A038164 * A193522 A187091 A034528

Adjacent sequences: A327379 A327380 A327381 * A327383 A327384 A327385

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved