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

0,4

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000041, A000120, A102437.

Sequence in context: A355573 A092991 A355649 * A162608 A143216 A086536

Adjacent sequences: A102422 A102423 A102424 * A102426 A102427 A102428

Keyword

nonn

Author

David S. Newman, Feb 23 2005

Status

approved