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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). 6
1, 1, 2, 3, 8, 18, 54, 155, 500, 1614, 5456, 18630, 64960, 228740, 814914, 2926323, 10589916, 38561814, 141219432, 519711666, 1921142832, 7129756188, 26555149404, 99228108222, 371886574632, 1397548389644, 5265131346368 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The Catalan bijection A057508 fixes only these kinds of trees, so this occurs in the table A073202 as row 168.
LINKS
FORMULA
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).
A073193(n) = (A000108(n) + A073192(n))/2.
MAPLE
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;
Cat := n -> binomial(2*n, n)/(n+1);
MATHEMATICA
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 *)
PROG
(PARI) Gat(n) = if (n == -1, 1, binomial(2*n, n)/(n+1));
a(n) = sum(i=0, n, if (!((n-i)%2), Gat((n-i)/2)*Gat(i-1))); \\ Michel Marcus, May 30 2018
CROSSREFS
Occurs for first time in A073202 as row 168.
Cf. also A073190.
Sequence in context: A185171 A339524 A158448 * A317722 A113183 A157015
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 25 2002
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)