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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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified July 2 13:15 EDT 2022. Contains 355007 sequences. (Running on oeis4.)