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A265846 Number of permutations of {1,2,...,2n} that result in a binary search tree of height n.

%I A265846 #32 May 14 2025 11:02:18
%S A265846 1,0,0,80,11360,1508032,197163648,27624399104,4256968553472,
%T A265846 724948555548672,136034167652859904,27976811931437752320,
%U A265846 6269253131872946298880,1522110257603797873737728,398311795891656532955725824,111813044381693547723191418880
%N A265846 Number of permutations of {1,2,...,2n} that result in a binary search tree of height n.
%H A265846 Alois P. Heinz, <a href="/A265846/b265846.txt">Table of n, a(n) for n = 0..250</a>
%H A265846 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_search_tree">Binary search tree</a>
%H A265846 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A265846 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A265846 a(n) = A195581(2n,n) = A244108(2n,n).
%F A265846 a(n) ~ c * d^n * (n-1)!, where d = -16*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 19.643259858273023595006139220961408..., c = 1/(2^(5/2)*Pi*sqrt(-1 - LambertW(-1, -exp(-1/2)/2))) = 0.0646979349409396546649864277836... . - _Vaclav Kotesovec_, Jan 02 2016, updated Mar 17 2024 and May 14 2025
%p A265846 b:= proc(n, k) option remember; `if`(n<2, `if`(k<n, 0, 1),
%p A265846       add(binomial(n-1, r)*b(r, k-1)*b(n-1-r, k-1), r=0..n-1))
%p A265846     end:
%p A265846 a:= n-> b(2*n, n)-b(2*n, n-1):
%p A265846 seq(a(n), n=0..20);
%t A265846 b[n_, k_] := b[n, k] = If[n<2, If[k<n, 0, 1], Sum[Binomial[n-1, r]*b[r, k-1]*b[n-1-r, k-1], {r, 0, n-1}]]; a[n_] := b[2*n, n] - b[2*n, n-1]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 25 2017, translated from Maple *)
%Y A265846 Cf. A195581, A244108.
%K A265846 nonn
%O A265846 0,4
%A A265846 _Alois P. Heinz_, Dec 21 2015