A100735 Inverse modulo 2 binomial transform of 2^n.

1, 1, 3, 3, 15, 15, 45, 45, 255, 255, 765, 765, 3825, 3825, 11475, 11475, 65535, 65535, 196605, 196605, 983025, 983025, 2949075, 2949075, 16711425, 16711425, 50134275, 50134275, 250671375, 250671375, 752014125, 752014125, 4294967295, 4294967295

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

Cf. A047999, A166282.

Sequence in context: A333862 A172087 A086116 * A362531 A129356 A290344

Adjacent sequences: A100732 A100733 A100734 * A100736 A100737 A100738

Keyword

easy,nonn

Author

Paul Barry, Dec 06 2004

Status

approved