A368746 Compositions (ordered partitions) of n into odd parts where the first part must be a maximal part.

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 18, 27, 40, 61, 93, 142, 217, 333, 512, 789, 1217, 1881, 2912, 4514, 7007, 10893, 16956, 26427, 41238, 64426, 100767, 157778, 247301, 388007, 609351, 957836, 1506928, 2372763, 3739035, 5896462, 9305388, 14695124, 23221657, 36718116, 58092690, 91961034

Offset: 0

Author

Joerg Arndt, Jan 04 2024

Keywords

• nonn

Examples

			The a(10) = 18 such compositions are:
   1:  [ 1 1 1 1 1 1 1 1 1 1 ]
   2:  [ 3 1 1 1 1 1 1 1 ]
   3:  [ 3 1 1 1 1 3 ]
   4:  [ 3 1 1 1 3 1 ]
   5:  [ 3 1 1 3 1 1 ]
   6:  [ 3 1 3 1 1 1 ]
   7:  [ 3 1 3 3 ]
   8:  [ 3 3 1 1 1 1 ]
   9:  [ 3 3 1 3 ]
  10:  [ 3 3 3 1 ]
  11:  [ 5 1 1 1 1 1 ]
  12:  [ 5 1 1 3 ]
  13:  [ 5 1 3 1 ]
  14:  [ 5 3 1 1 ]
  15:  [ 5 5 ]
  16:  [ 7 1 1 1 ]
  17:  [ 7 3 ]
  18:  [ 9 1 ]
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4801

Crossrefs

Cf. A079500.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1, `if`(m=0,
      add(b(n-2*j+1, 2*j-1), j=1..(n+1)/2), add(
      b(n-2*j+1, min(n-2*j+1, m)), j=1..(min(n, m)+1)/2)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 04 2024

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, 1, If[m == 0,
    Sum[b[n - 2j + 1, 2j - 1], {j, 1, (n + 1)/2}], Sum[
    b[n - 2j + 1, Min[n - 2j + 1, m]], {j, 1, (Min[n, m] + 1)/2}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 03 2024, after Alois P. Heinz *)

PARI

my(N=44, x='x+O('x^N)); Vec(1+sum(n=1, N, x^(2*n-1)/(1-sum(k=1, n, x^(2*k-1)))))

Formula

G.f.: 1 + Sum_{n>=1} x^(2*n-1)/(1 - Sum_{k=1..n} x^(2*k-1) ).