a(n)=sum_{j=0..n} sum_{i=0..j} F(i)*L(j), where F(n)=A000045(n) and L(n)=A000032(n).

Terms

a(0) =0a(1) =1a(2) =7a(3) =23a(4) =72a(5) =204a(6) =564a(7) =1521a(8) =4059a(9) =10747a(10) =28336a(11) =74504a(12) =195576a(13) =512865a(14) =1344063a(15) =3521007a(16) =9221688a(17) =24148468a(18) =63230860a(19) =165555665a(20) =433454835a(21) =1134839091a(22) =2971111392a(23) =7778574288a(24) =20364739632a(25) =53315851969a(26) =139583151799a(27) =365434146311a(28) =956720165544