A336273 Number of compositions of n containing at least one part p of multiplicity p.

0, 1, 0, 2, 3, 5, 14, 28, 44, 101, 207, 399, 779, 1609, 3122, 6121, 11804, 23631, 46273, 91604, 178096, 352419, 691996, 1371306, 2702206, 5356324, 10604748, 21080216, 41869930, 83383786, 166114046, 331434088, 661685588, 1322042390, 2642367028, 5283397304

Offset: 0

Author

Alois P. Heinz, Jul 15 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A011782, A336269.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> ceil(2^(n-1))-b(n$2, 0):
seq(a(n), n=0..40);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
     Sum[If[i == j, 0, b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := Ceiling[2^(n - 1)] - b[n, n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

Formula

a(n) = A011782(n) - A336269(n).