OFFSET
0,3
COMMENTS
In 1899, Glaisher, in particular, proved that the number of odd entries in the n-th row of Pascal triangle (A007318) is 2^s(n) (A001316), where s(n) is the number of 1's in the binary representation of n (s=A000120). Hence, all entries in the n-th row of Pascal triangle are odd if and only if n has the form 2^k-1, k=0,1,2,..., i.e., n is in A000225. In this case a(n) = OP(2^n) = 1. Thus the sequence contains infinitely many 1's.
REFERENCES
J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quarterly J. of Pure and Applied Math. 30 (1899), 150-156.
FORMULA
a(n) < 2^n.
EXAMPLE
{binomial(4,i)} = {1,4,6,4,1} -> {1,1,3,1,1}. Thus a(4)=1+1+3+1+1=7.
MATHEMATICA
oddPart[n_]:=n/2^IntegerExponent[n, 2];
Table[oddPart[Sum[oddPart[Binomial[n, i]], {i, 0, n}]], {n, 0, 50}] (* Peter J. C. Moses, Oct 27 2014 *)
PROG
(PARI) op(n) = n / 2^valuation(n, 2);
a(n) = op(sum(i=0, n, op(binomial(n, i)))); \\ Michel Marcus, Oct 31 2014
(Sage)
[odd_part(sum(odd_part(binomial(n, k)) for k in (0..n))) for n in range(37)] # Peter Luschny, Nov 05 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 27 2014
EXTENSIONS
More terms from Peter J. C. Moses, Oct 27 2014
STATUS
approved