The Lennon Sisters — Kathy, Janet, Peggy and Dianne — were a pretty popular act that appeared on The Lawrence Welk Show from, Wikipedia tells us, 1955 to 1968. After '68, they broke free from Mr. Welk and were around for a long time after that.
The person who posted the clip below on YouTube wrote "This song, and dance, was the opening for one of our shows; The Lennon Sisters Hour. Our choreographer, Jack Regas, worked very closely with our musical director, George Wyle, in staging these wonderful openings." I don't think it's from that show. The series the poster must be referencing was actually called Jimmy Durante Presents the Lennon Sisters Hour. It co-featured them with Guess Who and did involve Jack Regas and George Wyle.
But it was on ABC for one season — from 1969 to 1970 — and this is a medley of theme songs from later TV programs including Brian's Song (1971), Happy Days (which went on in '74), Baretta ('75) and Laverne & Shirley ('76). So I don't know where this is from…
When I came across this clip, it occurred to me that it probably inspired a segment on the Emmy Awards telecast for 1988. On it, The Sweeney Sisters from Saturday Night Live sang a not-dissimilar medley. That's Nora Dunn as Liz Sweeney and my lovely friend Jan Hooks as Candy Sweeney. Jan was a wonderful, talented lady who left us way too soon…