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A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

%I A055546 #61 Apr 18 2025 21:24:57
%S A055546 -1,2,-16,288,-9216,460800,-33177600,3251404800,-416179814400,
%T A055546 67421129932800,-13484225986560000,3263182688747520000,
%U A055546 -939796614359285760000,317651255653438586880000,-124519292216147926056960000,56033681497266566725632000000
%N A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.
%C A055546 Coefficient of the Cayley-Menger determinant of order n.
%C A055546 A roller coaster has n rows of seats, each of which has room for two people.  |a(n)| is the number of ways n men and n women can be seated with a man and a woman in each row. - _Geoffrey Critzer_, Dec 17 2011
%C A055546 The o.g.f. of 1/a(n) is -BesselI(0,i*sqrt(2*x)), with i the imaginary unit. See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - _Wolfdieter Lang_, Jan 10 2012
%C A055546 |a(n)|/2 is the number of integers k such that the digits of k and 2*k, written in base 2*n, are permutations of 0, 1, ..., 2*n-1. - _Yifan Xie_, Apr 12 2025
%H A055546 Andrew Howroyd, <a href="/A055546/b055546.txt">Table of n, a(n) for n = 0..100</a>
%H A055546 Paul C. Kainen, <a href="https://arxiv.org/abs/2302.13186">Construction numbers: How to build a graph?</a>, arXiv:2302.13186 [math.CO], 2023.
%H A055546 Usman A. Khan, Soummya Kar and Jose M. F. Moura, <a href="https://web.archive.org/web/20180412232727/http://www.eecs.tufts.edu/~khan/Courses/Spring2013/EE194/Lecs/LocalizationInWSNs_Khan.pdf">A novel geometric approach towards a linear theory for sensor localization</a>, 2013.
%H A055546 Alan L. Mackay, <a href="https://doi.org/10.1023/A:1019174403454">On the regular heptagon</a>, J. Math. Chemistry, vol. 21, 1997, 197-209.
%H A055546 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cayley-MengerDeterminant.html">Cayley-Menger Determinant</a>.
%F A055546 E.g.f.: -arcsinh(x/sqrt(2))^2. - _Vladeta Jovovic_, Aug 30 2004
%F A055546 Sum_{n>=0} |a(n)|/(2*n+1)! = Pi/2. - _Daniel Suteu_, Feb 06 2017
%F A055546 a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - _Terry D. Grant_, May 21 2017
%F A055546 From _Amiram Eldar_, Nov 18 2020: (Start)
%F A055546 Sum_{n>=0} 1/a(n) = (-1) * A334383.
%F A055546 Sum_{n>=0} (-1)^(n+1)/a(n) = A334381. (End)
%t A055546 Table[(-1)^(n+1)2^n n!^2, {n, 0, 20}]
%o A055546 (PARI) a(n)={(-1)^(n+1) * 2^n * n!^2} \\ _Andrew Howroyd_, Nov 07 2019
%Y A055546 Cf. A000079, A001044, A019669, A334381, A334383.
%Y A055546 Row of A340591 (in absolute values).
%K A055546 sign
%O A055546 0,2
%A A055546 _Eric W. Weisstein_
%E A055546 Terms a(14) and beyond from _Andrew Howroyd_, Nov 07 2019