This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227505 #11 Jul 19 2013 13:10:42 %S A227505 1,6,31,154,763,3808,19197,97772,502749,2607658,13630635,71743478, %T A227505 379949431,2023314980,10828048409,58206726936,314157742457, %U A227505 1701817879214,9249717805207,50427858276754,275695956722547,1511164724634440,8302888160922965 %N A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2). %C A227505 The terms of this sequence equal the Kn23 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows). %F A227505 a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)). %F A227505 a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2). %p A227505 A227505 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if k<n then return(0) fi; T(n, k-1)+T(n-1, k-1)+T(n-1, k) end; add(T(n-2*k+2,n-k+3), k=1..iquo(n+1, 2)) end: seq(A227505(n), n = 1..23); %p A227505 A227505 := proc(n): A006603(n+3) - A006318(n+3) - A006319(n+2) end: A006603 := n -> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): A006319 := proc(n): if n=0 then 1 else A006318(n) - A006318(n-1) fi: end: seq(A227505(n), n=1..23); %Y A227505 Cf. A033877, A006603, A006318, A006319, A227504. %K A227505 nonn,easy %O A227505 1,2 %A A227505 _Johannes W. Meijer_, Jul 15 2013