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%I #13 May 30 2018 13:52:11
%S 1,1,2,3,8,20,60,181,584,1916,6476,22210,77416,272840,971640,3488925,
%T 12621168,45946156,168206604,618853270,2286974856,8485246456,
%U 31596023208,118037654258,442287721872,1661790513944,6259494791096
%N Number of general plane trees which are either empty (the case a(0)), or whose root degree is either 1 (i.e., the planted trees) or the two leftmost subtrees (of the root node) are identical.
%C The Catalan bijection A072796 fixes only these kinds of trees, so this occurs in the table A073202 as row 1.
%H G. C. Greubel, <a href="/A073190/b073190.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0)=1, a(n) = Cat(n-1) + Sum_{i=0..n-2, (n-i) is even} Cat((n-i-2)/2)*Cat(i), where Cat(n) is A000108(n).
%p A073190 := proc(n) local d; Cat(n-1)+ add( (`mod`((n-d+1),2))*Cat((n-d-2)/2)*Cat(d), d=0..n-2); end;
%p Cat := n -> binomial(2*n,n)/(n+1);
%t a[n_] := CatalanNumber[n - 1] + Sum[Mod[n - d + 1, 2]*CatalanNumber[(n - d - 2)/2]*CatalanNumber[d], {d, 0, n - 2}]; a[0] = 1; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 06 2016 *)
%o (PARI) Cat(n) = binomial(2*n,n)/(n+1);
%o a(n) = if (n==0, 1, Cat(n-1) + sum(i=0, n-2, if (!((n-i)%2), Cat((n-i-2)/2)*Cat(i)))); \\ _Michel Marcus_, May 30 2018
%Y Occurs for first time in A073202 as row 1. A073191(n) = (A000108(n)+A073190(n))/2. Cf. also A073192.
%K nonn
%O 0,3
%A _Antti Karttunen_, Jun 25 2002
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