A327383 Number of colored integer partitions of n such that five colors are used and parts differ by size or by color.

1, 5, 20, 60, 160, 381, 845, 1760, 3495, 6660, 12267, 21935, 38230, 65140, 108785, 178437, 287975, 457965, 718575, 1113680, 1706533, 2587655, 3885615, 5781830, 8530625, 12486429, 18140360, 26169335, 37501595, 53403915, 75597130, 106408670, 148973260, 207496090

Offset

5,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 5..10000 (terms 5..5000 from Alois P. Heinz)

Wikipedia, Partition (number theory)

Formula

a(n) ~ exp(Pi*sqrt(5*n/3)) * 5^(1/4) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^5. - 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))(5):
seq(a(n), n=5..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 = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Crossrefs

Column k=5 of A308680.

Cf. A000009.

Sequence in context: A319888 A319869 A038165 * A339588 A344099 A215224

Adjacent sequences: A327380 A327381 A327382 * A327384 A327385 A327386

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved