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%I #12 Feb 12 2021 11:38:49
%S 1,1,2,5,14,50,194,1319,8834,117869,1269734,560616089,510535634,
%T 750085374614,2035030321602194,223759362505832069,764885974089796574,
%U 2887808734792025240864461034,716116205417604827135407034
%N Diagonal elements of the array in A341359.
%C a(n) plays a crucial role in the formula for the generating function for the n-th row of A341359. Essentially the terms A341359(n,k) for k >= n equals the halved coefficient of x^(k-n+1) in sqrt((1-(4*a(n)+1)*x)/(1-x)).
%C The sequence is not monotone. For example, a(11) > a(12).
%H T. Amdeberhan et al. <a href="https://mathoverflow.net/q/380482">Sequences generated by sum & product of terms (with rotating indices): combinatorial?</a>, 2021.
%t T[m_, 0] := 1; T[m_, n_] := T[m, n] = Sum[T[m, k] * T[m, Mod[n - 1 - k + m, n]], {k, 0, n - 1}]; Table[T[n, n], {n, 0, 18}] (* _Amiram Eldar_, Feb 09 2021 *)
%K nonn
%O 0,3
%A _Max Alekseyev_, Feb 09 2021