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A335063 a(n) = Sum_{k=0..n} (binomial(n,k) mod 2) * k. 2
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 (list; graph; refs; listen; history; text; internal format)
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}]

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 * A090546 A242901 A266013

Adjacent sequences:  A335060 A335061 A335062 * A335064 A335065 A335066

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 21 2020

STATUS

approved

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Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)