cp's OEIS Frontend

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.

%I A102425 #19 Jul 22 2018 20:36:23
%S A102425 1,0,1,2,2,4,6,6,12,16,20,28,40,48,69,91,111,150,197,238,319,398,493,
%T A102425 634,792,968,1226,1510,1846,2293,2811,3395,4197,5079,6126,7469,8993,
%U A102425 10781,13051,15593,18627,22333,26598,31571,37655,44569,52702,62462
%N 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.
%H A102425 Alois P. Heinz, <a href="/A102425/b102425.txt">Table of n, a(n) for n = 0..1000</a>
%e A102425 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.
%p A102425 p:= proc(n) option remember; local c, m;
%p A102425       c:= 0; m:= n;
%p A102425       while m>0 do c:= c +irem(m, 2, 'm') od;
%p A102425       c
%p A102425     end:
%p A102425 b:= proc(n,i,t) option remember;
%p A102425       if n<0 then 0
%p A102425     elif n=0 then 1-t
%p A102425     elif i=0 then 0
%p A102425     else b(n, i-1, t) +b(n-i, i, irem(p(i)+t, 2))
%p A102425       fi
%p A102425     end:
%p A102425 a:= n-> b(n, n, 0):
%p A102425 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 21 2011
%t A102425 Table[Length[Select[Map[Apply[Join,#]&,Map[IntegerDigits[#,2]&,Partitions[n]]],EvenQ[Count[#,1]]&]],{n,0,40}] (* _Geoffrey Critzer_, Sep 28 2013 *)
%o A102425 (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
%Y A102425 Cf. A000041, A000120, A102437.
%K A102425 nonn
%O A102425 0,4
%A A102425 _David S. Newman_, Feb 23 2005