A316996 - OEIS

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

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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 *)