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

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, 104, 92, 140, 162, 134, 217, 245, 207, 329, 343, 323, 489, 497, 489, 686, 726, 731, 980, 1040, 1036, 1400, 1477, 1491, 1970, 2038, 2139, 2744, 2835, 3016, 3752, 3939

Offset: 0

Author

Alois P. Heinz, Jul 24 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009, A000120, A316996.

Programs

Maple

h:= proc(n) option remember; `if`(n=0, 0, irem(n, 2, 'q')+h(q)) end:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..100);

Mathematica

h[n_] := h[n] = If[n == 0, 0, Mod[n, 2] + h[Quotient[n, 2]]];
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]]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 100](* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

Formula

a(n) + A316996(n) = A000009(n).

a(n) ~ exp(Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 09 2018