A320316 Number of weakly unimodal compositions of n in which the greatest part occurs exactly five times.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 13, 17, 24, 32, 44, 58, 79, 103, 138, 180, 237, 307, 402, 517, 670, 859, 1104, 1407, 1799, 2280, 2896, 3656, 4616, 5801, 7291, 9120, 11407, 14215, 17701, 21971, 27252, 33699, 41637, 51314, 63170, 77590, 95202

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=5 of A247255.

Programs

Maple

b:= proc(n, i) option remember; `if`(i>n, 0,
      `if`(5*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);

Mathematica

b[n_, i_] := b[n, i] = If[i > n, 0, If[5 i == n, 1, 0] +
     Sum[b[n - i j, i + 1] (j + 1), {j, 0, n/i}]];
a[n_] := If[n == 0, 1, b[n, 1]];
a /@ Range[0, 70] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

Formula

G.f.: Sum_{n>=0} x^(5*n) / Product_{j=1..n-1} (1-x^j)^2.

a(n) ~ Pi^4 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(13/4)). - Vaclav Kotesovec, Oct 24 2018