A327293 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size ten are used and the colors are introduced in increasing order.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 795, 1250, 1982, 3087, 4798, 7332, 11191, 16821, 25196, 37308, 54951, 80131, 117346, 169306, 244417, 349967, 500258, 709715, 1005550, 1414751, 1986544, 2773496, 3861747, 5349095, 7389698, 10178856, 13964050, 19102030

Offset: 55

Author

Alois P. Heinz, Aug 28 2019

Keywords

• nonn

Comments

In general, for k>=1, is column k of A321878 asymptotic to exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 55..5000

Crossrefs

Column k=10 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!)(10):
seq(a(n), n=55..93);

Formula

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-9))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-9)) / (4*10!*sqrt(30)*Pi*n). - Vaclav Kotesovec, Sep 18 2019