0,1

Or, a(n) = sum_{k >= 0} 2^wt(k) * binomial(wt(n+k),k).

G.f.: Prod_{ k >= 0 } (1 + 2*x^(2^k-1) + x^(2^k)).

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 09 2009: (Start)

Triangle begins:

.3;

.7,5;

.7,17,17,7;

.7,17,17,19,41,51,31,9;

.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,11;

.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,23,41,51,55,101,143,113,...

(End)

bin2:=proc(n, k) option remember; if k<0 or k>n then 0

elif k=0 then 1 else 2*bin2(n-1, k-1)+bin2(n-1, k); fi; end;

wt := proc(n) local w, m, i;

w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:

f:=n->add( bin2(wt(n+k), k), k=0..120 );

# or:

f := n->add( 2^k*binomial(wt(n+k), k), k=0..20 );

For generating functions of the form Prod_{k>=c} (1+a*x^(2^k-1)+b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694

Cf. A151689, A151691.

Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 09 2009]

Sequence in context: A116535 A122001 A161327 this_sequence A019809 A021270 A113910

Adjacent sequences: A151682 A151683 A151684 this_sequence A151686 A151687 A151688

nonn

N. J. A. Sloane (njas(AT)research.att.com), Jun 01 2009