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

15,2

Links

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

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

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 *)

Crossrefs

Column k=5 of A321878.

Sequence in context: A327290 A227356 A327289 * A102688 A236559 A275388

Adjacent sequences: A327285 A327286 A327287 * A327289 A327290 A327291

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2019

Status

approved