A227822 - OEIS

%I #25 Apr 02 2021 02:46:16

%S 1,1,4,220,60092152,203720181459953921762400,

%T 7088043372247785801830314829178419617696182324188730917543544992

%N Number of permutations of [n], [n+1], ... that result in a binary search tree of height n.

%C Empty external nodes are counted in determining the height of a search tree.

%H Alois P. Heinz, <a href="/A227822/b227822.txt">Table of n, a(n) for n = 0..9</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_search_tree">Binary search tree</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F a(n) = Sum_{k=n..2^n-1} A195581(k,n).

%e a(2) = 4, because 4 permutations of {1,2}, {1,2,3}, ... result in a binary search tree of height 2:

%e   (1,2):   1      (2,1):   2    (2,1,3), (2,3,1):    2

%e           / \             / \                      /   \

%e          o   2           1   o                    1     3

%e             / \         / \                      / \   / \

%e            o   o       o   o                    o   o o   o

%p b:= proc(n, k) option remember; `if`(n<2, `if`(k<n, 0, 1),

%p       add(binomial(n-1, r)*b(r, k-1)*b(n-1-r, k-1), r=0..n-1))

%p     end:

%p a:= n-> add(b(k, n)-b(k, n-1), k=n..2^n-1):

%p seq(a(n), n=0..6);

%t b[n_, k_] := b[n, k] = If[n < 2, If[k < n, 0, 1],

%t    Sum[Binomial[n - 1, r]*b[r, k - 1]*b[n - 1 - r, k - 1], {r, 0, n - 1}]];

%t a[n_] := Sum[b[k, n] - b[k, n - 1], {k, n, 2^n - 1}];

%t a /@ Range[0, 6] (* _Jean-François Alcover_, Apr 02 2021, after _Alois P. Heinz_ *)

%Y Column sums of A195581 and of A244108.

%Y Cf. A317012.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 31 2013

%E Terms corrected by _Alois P. Heinz_, Dec 08 2015