a(n) counts anti-chains of size two in "0,1,2" Motzkin trees on n edges.

Terms

a(0) =0a(1) =0a(2) =1a(3) =5a(4) =23a(5) =91a(6) =341a(7) =1221a(8) =4249a(9) =14465a(10) =48442a(11) =160134a(12) =523872a(13) =1699252a(14) =5472713a(15) =17520217a(16) =55801733a(17) =176942269a(18) =558906164a(19) =1759436704a(20) =5522119250a(21) =17285351782a(22) =53977433618a(23) =168194390290a(24) =523076690018