A320315 Number of weakly unimodal compositions of n in which the greatest part occurs exactly four times.

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 46, 62, 87, 116, 160, 212, 288, 380, 508, 666, 883, 1150, 1508, 1954, 2542, 3274, 4229, 5416, 6949, 8856, 11292, 14320, 18162, 22922, 28921, 36344, 45641, 57112, 71407, 88998, 110816, 137600, 170665

Offset

0,10

Links

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

Wikipedia, Unimodality, Unimodal function

Formula

G.f.: Sum_{n>=0} x^(4*n) / Product_{j=1..n-1} (1-x^j)^2.

a(n) ~ Pi^3 * exp(2*Pi*sqrt(n/3)) / (2^5 * 3^(5/4) * n^(11/4)). - Vaclav Kotesovec, Oct 24 2018

Maple

b:= proc(n, i) option remember; `if`(i>n, 0,
      `if`(4*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[4 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=4 of A247255.

Sequence in context: A064428 A052810 A364964 * A364956 A164090 A029744

Adjacent sequences: A320312 A320313 A320314 * A320316 A320317 A320318

Keyword

nonn

Author

Alois P. Heinz, Oct 10 2018

Status

approved