A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

1, 6, 31, 154, 763, 3808, 19197, 97772, 502749, 2607658, 13630635, 71743478, 379949431, 2023314980, 10828048409, 58206726936, 314157742457, 1701817879214, 9249717805207, 50427858276754, 275695956722547, 1511164724634440, 8302888160922965

Offset: 1

Author

Johannes W. Meijer, Jul 15 2013

Keywords

• nonn
• easy

Comments

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).

Crossrefs

Cf. A033877, A006603, A006318, A006319, A227504.

Programs

Maple

A227505 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227505(n), n = 1..23);
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);

Formula

a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)).

a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).