A348124 Number of compositions of n where the smallest part is smaller than the number of parts.

0, 1, 3, 6, 13, 28, 59, 122, 248, 501, 1009, 2028, 4070, 8159, 16343, 32717, 65472, 130991, 262041, 524157, 1048410, 2096943, 4194043, 8388285, 16776819, 33553946, 67108270, 134217002, 268434568, 536869825, 1073740493, 2147482019, 4294965305, 8589932164

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) + A098132(n) + A098133(n) = 2^(n-1).

Maple

b:= proc(n, s, c) option remember; `if`(s<c, ceil(2^(n-1)),
      `if`(n=0, 0, add(b(n-j, min(j, s), c+1), j=1..n)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 01 2021

Mathematica

b[n_, s_, c_] := b[n, s, c] = If[s < c, Ceiling[2^(n - 1)],
     If[n == 0, 0, Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

Crossrefs

Cf. A011782, A098132, A098133.

Sequence in context: A371564 A182137 A106461 * A095768 A002478 A106496

Adjacent sequences: A348121 A348122 A348123 * A348125 A348126 A348127

Keyword

nonn

Author

R. J. Mathar, Oct 01 2021

Extensions

a(23)-a(34) from Alois P. Heinz, Oct 01 2021

Status

approved