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%I #11 Jul 19 2013 13:10:42
%S 1,6,31,154,763,3808,19197,97772,502749,2607658,13630635,71743478,
%T 379949431,2023314980,10828048409,58206726936,314157742457,
%U 1701817879214,9249717805207,50427858276754,275695956722547,1511164724634440,8302888160922965
%N Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).
%C 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 a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)).
%F a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).
%p 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 := 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 Cf. A033877, A006603, A006318, A006319, A227504.
%K nonn,easy
%O 1,2
%A _Johannes W. Meijer_, Jul 15 2013