A102425 Let pi be an unrestricted partition of n with the summands written as binary numbers; a(n) is the number of such partitions with an even number of binary ones.

1, 0, 1, 2, 2, 4, 6, 6, 12, 16, 20, 28, 40, 48, 69, 91, 111, 150, 197, 238, 319, 398, 493, 634, 792, 968, 1226, 1510, 1846, 2293, 2811, 3395, 4197, 5079, 6126, 7469, 8993, 10781, 13051, 15593, 18627, 22333, 26598, 31571, 37655, 44569, 52702, 62462

Offset: 0

Author

David S. Newman, Feb 23 2005

Keywords

• nonn

Examples

			a(5) = 4 because there are 4 partitions of 5 whose binary representations have an even number of binary ones, namely 101, 100+1, 11+1+1, 10+1+1+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A000120, A102437.

Programs

Maple

p:= proc(n) option remember; local c, m;
      c:= 0; m:= n;
      while m>0 do c:= c +irem(m, 2, 'm') od;
      c
    end:
b:= proc(n,i,t) option remember;
      if n<0 then 0
    elif n=0 then 1-t
    elif i=0 then 0
    else b(n, i-1, t) +b(n-i, i, irem(p(i)+t, 2))
      fi
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

Mathematica

Table[Length[Select[Map[Apply[Join,#]&,Map[IntegerDigits[#,2]&,Partitions[n]]],EvenQ[Count[#,1]]&]],{n,0,40}] (* Geoffrey Critzer, Sep 28 2013 *)

PARI

seq(n)={apply(t->polcoeff(lift(t), 0), Vec(prod(i=1, n, 1/(1 - x^i*Mod( y^hammingweight(i), y^2-1 )) + O(x*x^n))))} \\ Andrew Howroyd, Jul 20 2018