Modulo 2 Binomial Transform

[Paul D. Hanna]

One last note regarding Paul Barry's Modulo 2 Binomial Transform. 
 
If we apply the triangle formed by reading Pascal's triangle mod 2: 
http://www.research.att.com/projects/OEIS?Anum=A047999 
as a transformation on a sequence A, and apply this transform M times, 
the formula for this (Mod 2 Binomial transformation)^M of A is: 
 
(1)  a(n) = Sum_{k=0..n} M^A000120(k)*(C(n,k) Mod 2)*A(k) 
where A000120(k) = number of 1's in binary expansion of k. 
 
This can be expressed using PARI code as:
a(n)=sum(k=0,n,M^subst(Pol(binary(n-k)),x,1)*(binomial(n,k)%2)*A(k))
 
  
A special case exists when A(n) = B^n for integer B; 
for then (1) is equivalent to the product formula: 
  
(2)  a(n) = Product_{k=0..[log_2(n)]} ( B^(2^k) + M )^b(n,k) 
where b(n,k) = coefficient of 2^k in the binary expansion of n. 
  
This can be expressed using PARI code as:
a(n)=prod(k=0,length(binary(n))-1,(B^(2^k)+M)^bittest(n,k)) 
  
The product formula (2) explains the factorization behavior noted
earlier. 
 
(Merry Christmas)^2,
      Paul
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