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a(n) = Sum_{j=0..n} C(n)*C(n-j), where C(n) is the n-th Catalan number.
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%I #7 Nov 16 2022 09:44:41

%S 1,2,8,45,322,2730,26004,268554,2940080,33635316,398300344,4849845000,

%T 60429982144,767721774800,9916427702880,129937069996965,

%U 1724052965464890,23129299114182030,313351935000465900,4282621342230699930,58994556159403576140,818487022124443918740

%N a(n) = Sum_{j=0..n} C(n)*C(n-j), where C(n) is the n-th Catalan number.

%F a(n) = C(n)*(C(n)*hypergeom([1, -n - 1], [1/2 - n], 1/4) + 1/2).

%F a(n) = ((-64*n^3 + 160*n^2 - 112*n + 24)*a(n-2) + (20*n^3 - 14*n^2 + 2*n)*a(n-1)) / (n*(n + 1)^2).

%p C := n -> binomial(2*n, n)/(n + 1):

%p A358436 := n -> add(C(n)*C(n-j), j = 0..n):

%p seq(A358436(n), n = 0..21);

%Y Cf. A000108, A358437.

%K nonn

%O 0,2

%A _Peter Luschny_, Nov 16 2022