A102437 Let pi be an unrestricted partition of n with the summands written in binary notation. a(n) is the number of such partitions whose binary representation has an odd number of binary ones.

0, 1, 1, 1, 3, 3, 5, 9, 10, 14, 22, 28, 37, 53, 66, 85, 120, 147, 188, 252, 308, 394, 509, 621, 783, 990, 1210, 1500, 1872, 2272, 2793, 3447, 4152, 5064, 6184, 7414, 8984, 10856, 12964, 15592, 18711, 22250, 26576, 31690, 37520, 44565, 52856, 62292

Offset

0,5

Links

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

Example

a(5) = 3 because there are 3 partitions of 5 with an odd number of binary ones in their binary representation, namely: 11+10, 10+10+1 and 1+1+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 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]]], OddQ[Count[#, 1]]&]], {n, 0, 40}] (* Geoffrey Critzer, Sep 28 2013 *)

Prog

(PARI) seq(n)={apply(t->polcoeff(lift(t), 1), 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, A102425, A316996.

Sequence in context: A136791 A213933 A091916 * A319794 A072706 A117433

Adjacent sequences: A102434 A102435 A102436 * A102438 A102439 A102440

Keyword

nonn

Author

David S. Newman, Feb 23 2005

Extensions

More terms from Vladeta Jovovic, Feb 23 2005

Status

approved