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

0, 1, 2, 6, 4, 10, 12, 28, 8, 18, 20, 44, 24, 52, 56, 120, 16, 34, 36, 76, 40, 84, 88, 184, 48, 100, 104, 216, 112, 232, 240, 496, 32, 66, 68, 140, 72, 148, 152, 312, 80, 164, 168, 344, 176, 360, 368, 752, 96, 196, 200, 408, 208, 424, 432, 880, 224, 456, 464, 944, 480

Offset

0,3

Comments

Modulo 2 binomial transform of nonnegative integers.

Links

Robert Israel, Table of n, a(n) for n = 0..3000

Formula

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

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

Maple

g:= proc(n, k) local L, M, t, j;
   L:= convert(k, base, 2);
   M:= convert(n, base, 2);
   1-max(zip(`*`, L, M))
end proc:
f:= n -> add(k*g(n-k, k), k=0..n):
map(f, [$0..100]); # Robert Israel, May 24 2020

Mathematica

Table[Sum[Mod[Binomial[n, k], 2] k, {k, 0, n}], {n, 0, 60}]
(* or *)
nmax = 60; CoefficientList[Series[(x/2) D[Product[(1 + 2 x^(2^k)), {k, 0, Log[2, nmax]}], x], {x, 0, nmax}], x]

Prog

(PARI) a(n) = n*2^(hammingweight(n)-1); \\ Michel Marcus, May 22 2020

Crossrefs

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

Sequence in context: A245788 A065879 A065880 * A371285 A090546 A242901

Adjacent sequences: A335060 A335061 A335062 * A335064 A335065 A335066

Keyword

nonn

Author

Ilya Gutkovskiy, May 21 2020

Status

approved