A335063 - OEIS

%I #17 Dec 05 2023 10:56:58

%S 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 48,100,104,216,112,232,240,496,32,66,68,140,72,148,152,312,80,164,

%U 168,344,176,360,368,752,96,196,200,408,208,424,432,880,224,456,464,944,480

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

%C Modulo 2 binomial transform of nonnegative integers.

%H Robert Israel, <a href="/A335063/b335063.txt">Table of n, a(n) for n = 0..3000</a>

%F G.f.: (x/2) * (d/dx) Product_{k>=0} (1 + 2 * x^(2^k)).

%F a(n) = n * 2^(A000120(n) - 1) = n * A001316(n) / 2.

%p g:= proc(n,k) local L,M,t,j;

%p    L:= convert(k,base,2);

%p    M:= convert(n,base,2);

%p    1-max(zip(`*`,L,M))

%p end proc:

%p f:= n -> add(k*g(n-k,k),k=0..n):

%p map(f, [$0..100]); # _Robert Israel_, May 24 2020

%t Table[Sum[Mod[Binomial[n, k], 2] k, {k, 0, n}], {n, 0, 60}]

%t (* or *)

%t nmax = 60; CoefficientList[Series[(x/2) D[Product[(1 + 2 x^(2^k)), {k, 0, Log[2, nmax]}], x], {x, 0, nmax}], x]

%o (PARI) a(n) = n*2^(hammingweight(n)-1); \\ _Michel Marcus_, May 22 2020

%Y Cf. A000120, A001316, A001787, A048298, A048896, A333176.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 21 2020