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A317239 Number of partitions of n into distinct parts with an even sum of Hamming weights.

%I A317239 #13 Dec 12 2020 14:16:18
%S A317239 1,0,0,2,0,2,3,0,4,6,3,6,9,6,12,19,9,19,31,17,37,44,29,62,68,55,91,
%T A317239 104,92,140,162,134,217,245,207,329,343,323,489,497,489,686,726,731,
%U A317239 980,1040,1036,1400,1477,1491,1970,2038,2139,2744,2835,3016,3752,3939
%N A317239 Number of partitions of n into distinct parts with an even sum of Hamming weights.
%H A317239 Alois P. Heinz, <a href="/A317239/b317239.txt">Table of n, a(n) for n = 0..10000</a>
%F A317239 a(n) + A316996(n) = A000009(n).
%F A317239 a(n) ~ exp(Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 09 2018
%p A317239 h:= proc(n) option remember; `if`(n=0, 0, irem(n, 2, 'q')+h(q)) end:
%p A317239 b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, t,
%p A317239       b(n, i-1, t)+b(n-i, min(n-i, i-1), irem(t+h(i), 2))))
%p A317239     end:
%p A317239 a:= n-> b(n$2, 1):
%p A317239 seq(a(n), n=0..100);
%t A317239 h[n_] := h[n] = If[n == 0, 0, Mod[n, 2] + h[Quotient[n, 2]]];
%t A317239 b[n_, i_, t_] := b[n, i, t] = If[i(i+1)/2 < n, 0, If[n == 0, t, b[n, i - 1, t] + b[n - i, Min[n - i, i - 1], Mod[t + h[i], 2]]]];
%t A317239 a[n_] := b[n, n, 1];
%t A317239 a /@ Range[0, 100](* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y A317239 Cf. A000009, A000120, A316996.
%K A317239 nonn
%O A317239 0,4
%A A317239 _Alois P. Heinz_, Jul 24 2018