aSEshaSayanam SEshaSayanam.....

aSEshaSayanam SEshaSayanam SEshaSaAyinaAm

SEshaAdriISaMaSEsham SriInivAsam BhajE~nisam

aSEshaSayanam...

Sayanam means sleeping/resting. Asesha means without remainder/entirely.

vaasanaad vaasudevasya vaasitaM te jagatrayam |
sarvabhuuta nivaaso asi vaasudeva namostu te ||

> aSEshaSayanam means…

the all pervading Supreme God VISHNU the VASUDEVA the one who rests in all beings/all creatures and there is no entity where He doesn't exist/rests.

SEshaSayanam...

SESHA refers to thousand headed Serpent God AadiSesha (Anantha) on whom Lord Vishnu reclines (Sayanam) - He is SeshaSayana; He is also known by other names BhujagaSayana; PannagaSayana; NaagaParyankaSayana; AnanthaSayana;

There is also another interpretation for SESHASAYANA; sEsha means remainder. At the time of great deluge when everything gets dissolved, the one and the only one entity that remains (Sesha)/exists is Lord Vishnu who is Achyuta who reclines on Banyan leaf when He is known as VATAPATRASAYEE/ VATAPATRASAYANA;

SEshaSaAyinaAm...

Sesha in this context refers to Lord Aadisesha the primordial Serpent God (Ananthaaya Namasthubhyam SahasraSirase Namaha) and SaAyee refers to the Couch (Sayana) and the Primordial Supreme God Vishnu is SESHASAAYEE the one who reclines on the serpent bed AadiSEsha.

SEshaAdriISaMaSEsham...

Eesha mean Lord; sEshaAdri refers to SesHachala Hills which is a manifestation of Lord AaDiSesha; sEshaAdrEEsha refers to Lord SrEenivaasa who is Lord of AaDiSesha who is sEshaAdri on the top of which the Lord is eternally stationed.

SrinivAsam BhajE~nisam...

I always pray to such Supreme God Srinivaasa (sEshaSaAyee) who is the Lord and consort of Goddess Lakshmi who always rests in His Hrudaya;

Above sloka is from AadityaPuranaAntargata Sri Venkatachala Mahatmya, where DevaSarma eulogizes Lord SriniVasa with the above sloka; 

Hari Sarvottama - Vaayu Jeevottama

Sri GuruRaajo Vijayate