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m = 100;
Product[If[EvenQ[DigitCount[k, 2, 1]], 1 + x^k, 1], {k, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Sep 11 2019, from PARI *)
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g:= proc(n) option remember; is(add(i, i=Bits[Split](n))::even) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
b(n, i-1)+`if`(g(i), b(n-i, min(n-i, i-1)), 0)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..100); # Alois P. Heinz, Sep 08 2019
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Andrew Howroyd, <a href="/A116491/b116491.txt">Table of n, a(n) for n = 0..1000</a>
(PARI) seq(n)=Vec(prod(k=1, n, if(hammingweight(k)%2==0, 1+x^k, 1) + O(x*x^n))) \\ Andrew Howroyd, Jun 22 2018
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