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

0,4

Links

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

Formula

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

Example

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 ]

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 *)

Prog

(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)))))

Crossrefs

Cf. A079500.

Sequence in context: A102543 A173383 A316076 * A325832 A068598 A293165

Adjacent sequences: A368743 A368744 A368745 * A368747 A368748 A368749

Keyword

nonn

Author

Joerg Arndt, Jan 04 2024

Status

approved