A327388 Number of colored integer partitions of n such that ten colors are used and parts differ by size or by color.

1, 10, 65, 320, 1320, 4762, 15500, 46410, 129710, 341990, 857695, 2059430, 4759235, 10630810, 23034880, 48562378, 99866045, 200766810, 395317950, 763661010, 1449390299, 2706189810, 4976391015, 9021860260, 16139848000, 28515535112, 49792637480, 85989053350

Offset: 10

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Comments

In general, column k > 0 of A308680 is asymptotic to exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..10000 (terms n = 5001..8000 from Vaclav Kotesovec)
• Wikipedia, Partition (number theory)

Crossrefs

Column k=10 of A308680.

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)*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))(10):
seq(a(n), n=10..45);

Mathematica

A327388[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^10, {x, 0, n}]; Table[A327388[n], {n, 10, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(10*n/3)) * 5^(1/4) / (2^(25/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^10. - Ilya Gutkovskiy, Jan 31 2021