a(n) is the sum of all peaks in the set of Catalan words of length n.

Terms

a(0) =0a(1) =0a(2) =0a(3) =1a(4) =6a(5) =27a(6) =111a(7) =441a(8) =1728a(9) =6733a(10) =26181a(11) =101763a(12) =395693a(13) =1539759a(14) =5997159a(15) =23381019a(16) =91244934a(17) =356427459a(18) =1393585779a(19) =5453514729a(20) =21358883439a(21) =83718027429a(22) =328380697629