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%I #63 Aug 02 2023 07:51:49
%S 1,1,2,80,21964800,74836825861835980800000,
%T 2606654998899867556195703676289609067340669424836280320000000000
%N Number of (binary) heaps on n levels (i.e., of 2^n - 1 elements).
%C A sequence {a_i}_{i=1..N} forms a (binary) heap if it satisfies a_i<a_{2i} and a_i<a_{2i+1} for 1<=i<=(N-1)/2.
%C a(n) is also the number of knockout tournament seedings that satisfy the increasing competitive intensity property. - _Alexander Karpov_, Aug 18 2015
%C a(n) is the number of coalescence sequences, or labeled histories, for a binary, leaf-labeled, rooted tree topology with 2^n leaves (Rosenberg 2006, Theorem 3.3 for a completely symmetric tree with 2^n leaves, dividing by Theorem 3.2 for 2^n leaves). - _Noah A Rosenberg_, Feb 12 2019
%C a(n+1) is also the number of random walk labelings of the perfect binary tree of height n, that begin at the root. - _Sela Fried_, Aug 02 2023
%H Alois P. Heinz, <a href="/A056972/b056972.txt">Table of n, a(n) for n = 0..9</a>
%H Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2308.00315">Random walk labelings of perfect trees and other graphs</a>, arXiv:2308.00315 [math.CO], 2023.
%H Alexander Karpov, <a href="http://www.uni-heidelberg.de/md/awi/forschung/dp600.pdf">A theory of knockout tournament seedings</a>, Heidelberg University, AWI Discussion Paper Series, No. 600.
%H Noah A. Rosenberg, <a href="https://doi.org/10.1007/s00026-006-0278-6">The Mean and Variance of the Numbers of r-Pronged Nodes and r-Caterpillars in Yule-Generated Genealogical Trees</a>, Ann. Combinator. 10 (2006), 129-146.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heap.html">Heap</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_heap">Binary heap</a>
%F a(n) = A056971(A000225(n)).
%F a(n) = A195581(2^n-1,n).
%F a(n) = binomial(2^n-2, 2^(n-1)-1)*a(n-1)^2. - _Robert Israel_, Aug 18 2015, from the Mathematica program
%F a(n) = (2^n-1)!/Product_{k=1..n} (2^k-1)^(2^(n-k)). - _Robert Israel_, Aug 18 2015, from the Maple program
%e There is 1 heap on 2^0-1=0 elements, 1 heap on 2^1-1=1 element and there are 2 heaps on 2^2-1=3 elements and so on.
%p a:= n-> (2^n-1)!/mul((2^k-1)^(2^(n-k)), k=1..n):
%p seq(a(i), i=0..6); # _Alois P. Heinz_, Nov 22 2007
%t s[1] := 1; s[l_] := s[l] := Binomial[2^l-2, 2^(l-1)-1]s[l-1]^2; Table[s[l], {l, 10}]
%Y Cf. A000225, A056971, A195581.
%Y Column k=2 of A273712.
%K nonn
%O 0,3
%A _Eric W. Weisstein_
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