%I #22 Apr 26 2022 02:58:40

%S 0,1,7,97,6031,8760337,8245932762607,3508518207942911995940881,

%T 311594265746788494170059418351454897488270152687

%N Total number of internal nodes in all binary search trees of height n.

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

%H Alois P. Heinz, <a href="/A335922/b335922.txt">Table of n, a(n) for n = 0..12</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} k * A335919(k,n) = Sum_{k=n..2^n-1} k * A335920(k,n).

%e a(2) = 7 = 2 + 3 + 2:

%e .

%e 2 2 1

%e / \ / \ / \

%e 1 o 1 3 o 2

%e / \ ( ) ( ) / \

%e o o o o o o o o

%e .

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

%p add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))

%p end:

%p T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):

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

%p seq(a(n), n=0..10);

%t b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,

%t Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];

%t T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];

%t a[k_] := Sum[T[n, k]*n, {n, k, 2^k - 1}];

%t Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Apr 26 2022, after _Alois P. Heinz_ *)

%Y Cf. A317012, A335919, A335920, A335921.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jun 29 2020