|
|
A073190
|
|
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.
|
|
10
|
|
|
1, 1, 2, 3, 8, 20, 60, 181, 584, 1916, 6476, 22210, 77416, 272840, 971640, 3488925, 12621168, 45946156, 168206604, 618853270, 2286974856, 8485246456, 31596023208, 118037654258, 442287721872, 1661790513944, 6259494791096
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The Catalan bijection A072796 fixes only these kinds of trees, so this occurs in the table A073202 as row 1.
|
|
LINKS
|
|
|
FORMULA
|
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).
|
|
MAPLE
|
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;
Cat := n -> binomial(2*n, n)/(n+1);
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(PARI) Cat(n) = binomial(2*n, n)/(n+1);
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|