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XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.
2

%I #15 Aug 11 2024 16:46:27

%S 1,2,4,0,8,0,0,0,16,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,64,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,128

%N XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.

%H Harvey P. Dale, <a href="/A099890/b099890.txt">Table of n, a(n) for n = 0..1024</a>

%F a(0) = 1; a(2^n) = 2^(n+1); a(n) = 0 otherwise.

%t Join[{1},Table[If[IntegerQ[Log2[n]],2n,0],{n,512}]] (* or *) Join[{1},With[{kk=10},Flatten[ Riffle[ 2^Range[kk],Table[PadRight[{},2^k-1,0],{k,0,kk}]]]]] (* _Harvey P. Dale_, Aug 11 2024 *)

%o (PARI) a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*(2*(n-i)+1)));B

%Y Cf. A099884, A099889.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 29 2004