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A316996 Consider a partition of n into distinct parts 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.

%I A316996 #27 Sep 16 2019 08:23:51
%S A316996 0,1,1,0,2,1,1,5,2,2,7,6,6,12,10,8,23,19,15,37,27,32,60,42,54,87,74,
%T A316996 88,130,116,134,206,173,203,305,256,325,437,375,485,624,574,700,879,
%U A316996 836,1008,1268,1190,1433,1773,1688,2059,2443,2376,2883,3362,3356,3978
%N A316996 Consider a partition of n into distinct parts 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.
%H A316996 Alois P. Heinz, <a href="/A316996/b316996.txt">Table of n, a(n) for n = 0..10000</a>
%F A316996 a(n) + A317239(n) = A000009(n).
%F A316996 a(n) ~ exp(Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 09 2018
%e A316996 For n = 5 there are 3 partitions to be examined: 5, 4+1, and 3+2. In binary these are 101, 100+1, and 11+10, which have 2, 2, and 3 binary ones respectively, so a(5) = 1.
%p A316996 h:= proc(n) option remember; `if`(n=0, 0, irem(n, 2, 'q')+h(q)) end:
%p A316996 b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, t,
%p A316996       b(n, i-1, t)+b(n-i, min(n-i, i-1), irem(t+h(i), 2))))
%p A316996     end:
%p A316996 a:= n-> b(n$2, 0):
%p A316996 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 24 2018
%t A316996 h[n_] := h[n] = If[n == 0, 0, Mod[n, 2] + h[Quotient[n, 2]]];
%t A316996 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 A316996 a[n_] := b[n, n, 0];
%t A316996 a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ *)
%o A316996 (PARI) seq(n)={apply(t->polcoeff(lift(t), 1), Vec(prod(i=1, n, 1 + x^i*Mod( y^hammingweight(i), y^2-1 ) + O(x*x^n))))} \\ _Andrew Howroyd_, Jul 23 2018
%Y A316996 Cf. A000009, A000120, A102437, A317239.
%K A316996 nonn
%O A316996 0,5
%A A316996 _David S. Newman_, Jul 18 2018