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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.
2
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
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
KEYWORD
nonn
AUTHOR
David S. Newman, Feb 23 2005
STATUS
approved