A320319 Number of weakly unimodal compositions of n in which the greatest part occurs exactly eight times.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 16, 20, 27, 34, 45, 56, 73, 90, 115, 142, 179, 220, 276, 338, 421, 516, 638, 780, 962, 1172, 1438, 1750, 2138, 2594, 3159, 3822, 4638, 5600, 6771, 8154, 9834, 11812, 14203, 17024

Offset: 0

Author

Alois P. Heinz, Oct 10 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Wikipedia, Unimodality, Unimodal function

Crossrefs

Column k=8 of A247255.

Programs

Maple

b:= proc(n, i) option remember; `if`(i>n, 0,
      `if`(8*i=n, 1, 0)+add(b(n-i*j, i+1)*(j+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..70);

Formula

G.f.: Sum_{n>=0} x^(8*n) / Product_{j=1..n-1} (1-x^j)^2.

a(n) ~ Pi^7 * 7! * exp(2*Pi*sqrt(n/3)) / (2^10 * 3^(17/4) * n^(19/4)). - Vaclav Kotesovec, Oct 24 2018