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.

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, 88, 130, 116, 134, 206, 173, 203, 305, 256, 325, 437, 375, 485, 624, 574, 700, 879, 836, 1008, 1268, 1190, 1433, 1773, 1688, 2059, 2443, 2376, 2883, 3362, 3356, 3978

Offset

0,5

Links

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

Formula

a(n) + A317239(n) = A000009(n).

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

Example

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.

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<n, 0, `if`(n=0, t,
      b(n, i-1, t)+b(n-i, min(n-i, i-1), irem(t+h(i), 2))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 24 2018

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, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)

Prog

(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

Crossrefs

Cf. A000009, A000120, A102437, A317239.

Sequence in context: A075259 A307877 A259703 * A169589 A003570 A011281

Adjacent sequences: A316993 A316994 A316995 * A316997 A316998 A316999

Keyword

nonn

Author

David S. Newman, Jul 18 2018

Status

approved