A258259 Number of partitions of n into distinct parts less than or equal to n/2.

1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 4, 8, 8, 13, 13, 21, 21, 31, 33, 46, 49, 67, 72, 95, 104, 134, 146, 186, 203, 253, 279, 343, 378, 461, 507, 611, 675, 806, 889, 1055, 1163, 1369, 1512, 1768, 1950, 2270, 2502, 2896, 3193, 3678, 4051, 4649, 5117, 5847

Offset: 0

Author

Geoffrey Critzer, May 24 2015

Keywords

• nonn

Comments

Intuitively the sequence is asymptotic to A000009. a(300)/A000009(300) is approximately .997749.

Examples

			a(9) = 1 because we have: 2+3+4.
a(10) = 3 because we have: 1+4+5, 2+3+5, 1+2+3+4.
		

Links

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

Crossrefs

Cf. A000009.

Programs

Maple

b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
    end:
a:= n-> b(n, iquo(n, 2)):
seq(a(n), n=0..60);  # Alois P. Heinz, May 25 2015

Mathematica

Prepend[Table[nn = n;Coefficient[Series[Product[1 + x^i, {i, 1, nn/2}], {x, 0, nn}],x^n], {n, 1, 50}], 1]

Formula

a(n) = [x^n] Product_{i=1..floor(n/2)} 1 + x^i.