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A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).

%I A073192 #19 Sep 09 2023 19:19:58
%S A073192 1,1,2,3,8,18,54,155,500,1614,5456,18630,64960,228740,814914,2926323,
%T A073192 10589916,38561814,141219432,519711666,1921142832,7129756188,
%U A073192 26555149404,99228108222,371886574632,1397548389644,5265131346368
%N A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).
%C A073192 The Catalan bijection A057508 fixes only these kinds of trees, so this occurs in the table A073202 as row 168.
%H A073192 G. C. Greubel, <a href="/A073192/b073192.txt">Table of n, a(n) for n = 0..1000</a>
%F A073192 a(n) = Sum_{i=0..n, (n-i) is even} Gat((n-i)/2)*Gat(i-1), where Gat(-1) = 1 and otherwise like A000108(n).
%F A073192 A073193(n) = (A000108(n) + A073192(n))/2.
%p A073192 A073192 := proc(n) local d; add( (`mod`((n-d+1),2))*Cat((n-d)/2)*(`if`((0=d),1,Cat(d-1))), d=0..n); end;
%p A073192 Cat := n -> binomial(2*n,n)/(n+1);
%t A073192 a[n_] := Sum[Mod[n - k + 1, 2]*CatalanNumber[(n - k)/2]*If[k == 0, 1, CatalanNumber[k - 1]], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 05 2016 *)
%o A073192 (PARI) Gat(n) = if (n == -1, 1, binomial(2*n,n)/(n+1));
%o A073192 a(n) = sum(i=0, n, if (!((n-i)%2), Gat((n-i)/2)*Gat(i-1))); \\ _Michel Marcus_, May 30 2018
%Y A073192 Occurs for first time in A073202 as row 168.
%Y A073192 Cf. also A073190.
%K A073192 nonn
%O A073192 0,3
%A A073192 _Antti Karttunen_, Jun 25 2002