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

0,12

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Wikipedia, Unimodality, Unimodal function

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

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 *)

Crossrefs

Column k=5 of A247255.

Sequence in context: A055167 A064628 A188674 * A236166 A017834 A286225

Adjacent sequences: A320313 A320314 A320315 * A320317 A320318 A320319

Keyword

nonn

Author

Alois P. Heinz, Oct 10 2018

Status

approved