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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.
3
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
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
KEYWORD
nonn
AUTHOR
David S. Newman, Feb 23 2005
EXTENSIONS
More terms from Vladeta Jovovic, Feb 23 2005
STATUS
approved