A327288 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.

1, 2, 5, 10, 20, 36, 73, 125, 222, 372, 623, 1002, 1611, 2559, 3984, 6139, 9355, 14096, 21028, 31093, 45523, 66403, 95779, 137495, 195813, 277531, 390428, 546942, 761113, 1054749, 1454412, 1996271, 2727247, 3711683, 5029288, 6789347, 9130315, 12234596, 16335987

Offset: 15

Author

Alois P. Heinz, Aug 28 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 15..5000

Crossrefs

Column k=5 of A321878.

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))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(5):
seq(a(n), n=15..53);

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]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[15, 53] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-4))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-4)) / (4*5!*sqrt(15)*Pi*n). - Vaclav Kotesovec, Sep 18 2019