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%I #15 Jul 19 2013 13:09:53
%S 1,4,17,74,335,1566,7515,36836,183709,929392,4758477,24611950,
%T 128411643,675051770,3572165431,19012868648,101718917721,546707554844,
%U 2950563205705,15983712882930,86880753686279,473710078493718,2590187432233363,14199709022579788
%N Schroeder triangle sums: a(n) = A006603(n+1) - A006318(n+1).
%C The terms of this sequence equal the Kn22 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+2), k=1..floor((n+1)/2)).
%F a(n) = A006603(n+1) - A006318(n+1).
%p A227504 := 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+2), k=1..iquo(n+1, 2)) end: seq(A227504(n), n = 1..24);
%p A227504 := proc(n): A006603(n+1) - A006318(n+1) 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): seq(A227504(n), n=1..24);
%Y Cf. A033877, A006603, A006318, A227505.
%K nonn,easy
%O 1,2
%A _Johannes W. Meijer_, Jul 15 2013
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