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A335063 a(n) = Sum_{k=0..n} (binomial(n,k) mod 2) * k.

%I A335063 #17 Dec 05 2023 10:56:58
%S A335063 0,1,2,6,4,10,12,28,8,18,20,44,24,52,56,120,16,34,36,76,40,84,88,184,
%T A335063 48,100,104,216,112,232,240,496,32,66,68,140,72,148,152,312,80,164,
%U A335063 168,344,176,360,368,752,96,196,200,408,208,424,432,880,224,456,464,944,480
%N A335063 a(n) = Sum_{k=0..n} (binomial(n,k) mod 2) * k.
%C A335063 Modulo 2 binomial transform of nonnegative integers.
%H A335063 Robert Israel, <a href="/A335063/b335063.txt">Table of n, a(n) for n = 0..3000</a>
%F A335063 G.f.: (x/2) * (d/dx) Product_{k>=0} (1 + 2 * x^(2^k)).
%F A335063 a(n) = n * 2^(A000120(n) - 1) = n * A001316(n) / 2.
%p A335063 g:= proc(n,k) local L,M,t,j;
%p A335063    L:= convert(k,base,2);
%p A335063    M:= convert(n,base,2);
%p A335063    1-max(zip(`*`,L,M))
%p A335063 end proc:
%p A335063 f:= n -> add(k*g(n-k,k),k=0..n):
%p A335063 map(f, [$0..100]); # _Robert Israel_, May 24 2020
%t A335063 Table[Sum[Mod[Binomial[n, k], 2] k, {k, 0, n}], {n, 0, 60}]
%t A335063 (* or *)
%t A335063 nmax = 60; CoefficientList[Series[(x/2) D[Product[(1 + 2 x^(2^k)), {k, 0, Log[2, nmax]}], x], {x, 0, nmax}], x]
%o A335063 (PARI) a(n) = n*2^(hammingweight(n)-1); \\ _Michel Marcus_, May 22 2020
%Y A335063 Cf. A000120, A001316, A001787, A048298, A048896, A333176.
%K A335063 nonn
%O A335063 0,3
%A A335063 _Ilya Gutkovskiy_, May 21 2020