A B737 FLT CREW RECEIVE A TCASII RA FOR AN NMAC JUST AFTER DEPARTING SEA WITH IFR TFC DEPARTING OFF PARALLEL RWY.

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INITIAL CLRNC: SEATTLE 3; MAINTAIN 9000 FT. WE WERE #3 IN LINE ON TXWY B BEHIND 2 ACFT. WHILE THE OTHER 2 ACFT WERE HOLDING SHORT OF RWY 16L; WE WERE TOLD TO TAXI ACROSS RWY 16L AND CLRED FOR TKOF ON RWY 16R. WE WERE TOLD THAT WE COULD EXPECT AN EARLY R TURN AND THAT THE OTHER TFC ON RWY 16L WOULD BE STRAIGHT OUT. WE CONDUCTED A NORMAL ROLLING TKOF ON RWY 16R. AT APPROX 1000 FT MSL; WE RECEIVED A TCASII RA OF 'DSND; DSND; DSND.' I PUSHED THE NOSE OVER LEVELING THE ACFT WHILE LOOKING FOR THE CONFLICTING TFC. THE FIRST AND ONLY TIME I SAW THE OTHER ACFT WAS AT THIS POINT. HE WAS ABOUT 400 FT TO OUR L AND 75 FT BELOW US (8 O'CLOCK LOW POS). CONCENTRATING ON FLYING THE JET; I LOST SIGHT OF ACFT (BEHIND AND BELOW US). AT APPROX 3 MI ON DEP; WE WERE SWITCHED OVER TO DEP CTL; WHERE WE WERE ASKED IF WE HAD VISUAL CONTACT WITH ACFT Y. WE REPLIED 'NEGATIVE.' WHEN ACFT Y CHKED IN WITH DEP; THEY WERE ASKED THE SAME QUESTION. THEY ALSO REPLIED 'NEGATIVE.' OBSERVATION OF THE ACFT ALT RPTING SHOWED ACFT Y DIRECTLY ABOVE US WITH 400 FT OF SEPARATION. DEP CTL ISSUED US A R TURN; WHILE ACFT Y CONTINUED STRAIGHT AHEAD.