A053197 Number of level partitions of n.

1, 1, 2, 2, 4, 3, 6, 5, 10, 8, 13, 12, 21, 18, 27, 27, 42, 38, 54, 54, 77, 76, 101, 104, 143, 142, 183, 192, 249, 256, 323, 340, 432, 448, 550, 585, 722, 760, 918, 982, 1190, 1260, 1502, 1610, 1917, 2048, 2408, 2590, 3053, 3264, 3800, 4097, 4765, 5120, 5910, 6378

Offset: 0

Author

Vladeta Jovovic, Mar 02 2000

Keywords

• nonn
• nice

Comments

A partition is level if the powers of 2 dividing its parts are all equal.

Links

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

Crossrefs

Cf. A049313, A053195.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-p, p), j=0..n/i)))
    end:
a:= n-> (m-> `if`(n=0, 1, add(b(n, (h-> h-1+irem(h, 2)
    )(iquo(n, 2^j))*2^j, 2^(1+j)), j=0..m)))(ilog2(n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 11 2015

Mathematica

a[n_] := Sum[ PartitionsQ[n/2^k], {k, 0, IntegerExponent[n, 2]}]; Table[ a[n], {n, 1, 55}] (* Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)

Formula

a(n) = Sum_{k=0..A007814(n)} A000009(n/2^k). a(2*n+1) = A000009(2*n+1) = A078408(n). - Vladeta Jovovic, Sep 29 2004

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 11 2015