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A344500 a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121. 1

%I

%S 1,1,2,7,21,66,216,715,2395,8101,27598,94568,325612,1125632,3904512,

%T 13583195,47373255,165585883,579907758,2034443127,7148313381,

%U 25151582046,88607951512,312518438532,1103393962996,3899415207676,13792683831176,48825746365672,172971084083752

%N a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.

%F a(n) = Sum_{j=0..n} even(n + j)*binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1), where even(k) = 1 if k is even and otherwise 0.

%p a := n -> add((if n + j mod 2 = 1 then 0 else binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1) fi), j = 0..n): seq(a(n), n = 0..28);

%Y Cf. A053121, A344501.

%K nonn

%O 0,3

%A _Peter Luschny_, May 22 2021

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Last modified January 27 03:09 EST 2022. Contains 350601 sequences. (Running on oeis4.)