OFFSET
0,3
COMMENTS
The modulo 2 binomial transform and its inverse are defined by
B(n) = Sum_{k=0..n} (binomial(n,k) mod 2)*A(k),
A(n) = Sum_{k=0..n} (-1)^A010060(n-k)*(binomial(n, k) mod 2)*B(k). - N. J. A. Sloane, Dec 20 2019
2^n may be retrieved as Sum_{k=0..n} mod(binomial(n,k),2)*a(k).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Thomas Baruchel, A non-symmetric divide-and-conquer recursive formula for the convolution of polynomials and power series, arXiv:1912.00452 [math.NT], 2019.
FORMULA
a(n) = Sum_{k=0..n} (-1)^A010060(n-k)*mod(binomial(n, k), 2)*2^k.
MATHEMATICA
Table[Sum[(-1)^ThueMorse[n - k]*Mod[Binomial[n, k], 2]*2^k, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Apr 17 2018 *)
PROG
(PARI) for(n=0, 50, print1(abs(sum(k=0, n, (-1)^(hammingweight(k)%2)* lift(Mod(binomial(n, k), 2))*2^k)), ", ")) \\ G. C. Greubel, Apr 17 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 06 2004
STATUS
approved