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A238452 Second column of the extended Catalan triangle A189231.

%I A238452 #19 Jun 07 2021 13:31:19
%S A238452 0,1,2,2,8,5,30,14,112,42,420,132,1584,429,6006,1430,22880,4862,87516,
%T A238452 16796,335920,58786,1293292,208012,4992288,742900,19315400,2674440,
%U A238452 74884320,9694845,290845350,35357670,1131445440,129644790,4407922860,477638700,17194993200
%N A238452 Second column of the extended Catalan triangle A189231.
%F A238452 Definition: a(n) = binomial(n+1, floor(n/2)+1) / (floor(n/2)+2) if n is odd, and 2*binomial(n, floor(n/2)+1) otherwise.
%F A238452 a(n) = A189231(n, 1).
%F A238452 a(n) = A238762(n+1, n-1).
%F A238452 a(2*n) = A162551(n).
%F A238452 a(2*n+1) = A000108(n+1).
%F A238452 a(n) = A057977(n+1) - A057977(n)*((n+1) mod 2). - _Peter Luschny_, Aug 07 2016
%p A238452 a := proc(n) option remember;
%p A238452   if n < 3 then return n fi;
%p A238452   if n mod 2 = 0 then return n*a(n-1) fi;
%p A238452   h := iquo(n,2); n*a(n-1)/(h*(h+2)) end:
%p A238452 seq(a(n), n=0..36);
%t A238452 t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] =
%t A238452   t[n - 1, k - 1] + Mod[n - k, 2] t[n - 1, k] + t[n - 1, k + 1];
%t A238452 a[n_] := t[n, 1];
%t A238452 Table[a[n], {n, 0, 36}] (* _Jean-François Alcover_, Jul 10 2019 *)
%o A238452 (Sage)
%o A238452 def A238452():
%o A238452     a = 1; n = 2
%o A238452     yield 0
%o A238452     while True:
%o A238452         yield a
%o A238452         a *= n
%o A238452         if is_odd(n):
%o A238452             a /= (n//2*(n//2+2))
%o A238452         n += 1
%o A238452 a = A238452(); [next(a) for n in range(36)]
%Y A238452 Cf. A000108, A057977, A162551, A189231.
%K A238452 nonn,easy
%O A238452 0,3
%A A238452 _Peter Luschny_, Mar 01 2014