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

Terms

a(0) =0a(1) =0a(2) =0a(3) =0a(4) =1a(5) =8a(6) =44a(7) =209a(8) =924a(9) =3927a(10) =16303a(11) =66691a(12) =270181a(13) =1087371a(14) =4356131a(15) =17394026a(16) =69289961a(17) =275543036a(18) =1094352236a(19) =4342295396a(20) =17218070066a(21) =68239187876a(22) =270351828476