ZOB CTLR EXPERIENCED OPERROR AT FL350 AS 2 CONFLICTING ACFT WERE XFERRED TO ADJACENT SECTORS AND TIMELY COORD WAS NOT AFFECTED.

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LR60 AT FL390 WBOUND LNDG PTK. B757 ON J36 WBOUND ENRTE CRUISE. DSNDED LR60 TO FL350 AND VECTORED N FOR ADDITIONAL SPACING WITH OTHER LNDG TFC AND TO STAY N OF B757. RECLRED LR60 TO YQO; EFFECTED HDOF AND CHANGED FREQ. NEW CLRNC TO LR60 ENSURED SEPARATION WITH B757 RECEIVING CTLR ASKED ME TO PUT LR60 ON 240 DEG HDG AND I INFORMED HIM I'D ALREADY XFERRED COMS AND THAT LR60 WOULD BE HIS CTL. RECEIVING CTLR TURNED LR60 TO A 210 DEG HDG AND DSNDED TO FL330. I OBSERVED LR60 LEAVE FL350 AND AT THAT TIME XFERRED COMS OF B757 TO THE NEXT SECTOR W. I FELT THERE WAS MORE THAN ADEQUATE ROOM BTWN THE TWO. LR60 DSNDED VERY SLOWLY OUT OF FL345 AND CONFLICT ALERT ACTIVATED. I CALLED THE LR60 CTLR TO TELL HIM TO EXPEDITE THE LR60 DSCNT. HE HAD ALREADY XFERRED LR60 COMS TO THE NEXT SECTOR AND AN 'EXPEDITE DSCNT' CLRNC WAS NEVER ISSUED. I HAD THE B757 RECEIVING CTLR VECTOR HIS ACFT AWAY FROM THE TFC; BUT TOO LATE TO AVOID LOSS OF SEPARATION. I SHOULD HAVE POINTED OUT THE B757 TO THE FIRST CTLR WORKING THE LR60; TOLD HIM TO VECTOR THE LR60 N INSTEAD OF S; OR KEPT THE B757 ON MY FREQ UNTIL THERE WAS NO POSSIBILITY OF CONFLICT. AVOIDING THE DELAY IN COORDINATING THE SOUTHERLY TURN OF THE B757 MIGHT HAVE AFFORDED ENOUGH TIME TO PROVIDE SEPARATION. SUPPLEMENTAL INFO FROM ACN 640865: I RECEIVED HDOF ON THE LEAR DSNDING TO FL330 ON A HDG OF 210 DEGS. ACFT CHKED ON IMMEDIATELY AND STARTED TO FLASH WITH ACFT WBOUND (B757) AT FL350. I ACKNOWLEDGED THE LEAR AND TURNED HIM W AND TOLD HIM TO RPT LEVEL AT FL330. AT TIME OF CONTACT; THE ACFT WERE ABOUT 5 MI AND CLOSING (WINDS FROM N). CONFLICT ALERT WAS 4.1 MI AND 1300 FT VERT. I OWNED FL330 TO FL240. ACFT WAS A FEW MI OUTSIDE MY AIRSPACE AT TIME OF HDOF. I KNEW NOTHING ABOUT THE B757 UNTIL THE CONFLICT ALERT FLASHED; SINCE THAT IS ABOVE MY AIRSPACE. I ACTED AS QUICK AS I COULD TO KEEP ACFT FROM HITTING. THIS INVOLVED 4 SECTORS.