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a(n) = (-1)^(n+1) * 2^n * n!^2.
7

%I #57 Mar 07 2023 02:37:23

%S -1,2,-16,288,-9216,460800,-33177600,3251404800,-416179814400,

%T 67421129932800,-13484225986560000,3263182688747520000,

%U -939796614359285760000,317651255653438586880000,-124519292216147926056960000,56033681497266566725632000000

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

%C Coefficient of the Cayley-Menger determinant of order n.

%C 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 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

%H Andrew Howroyd, <a href="/A055546/b055546.txt">Table of n, a(n) for n = 0..100</a>

%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cayley-MengerDeterminant.html">Cayley-Menger Determinant</a>.

%F E.g.f.: -arcsinh(x/sqrt(2))^2. - _Vladeta Jovovic_, Aug 30 2004

%F Sum_{n>=0} |a(n)|/(2*n+1)! = Pi/2. - _Daniel Suteu_, Feb 06 2017

%F a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - _Terry D. Grant_, May 21 2017

%F From _Amiram Eldar_, Nov 18 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (-1) * A334383.

%F Sum_{n>=0} (-1)^(n+1)/a(n) = A334381. (End)

%t Table[(-1)^(n+1)2^n n!^2, {n, 0, 20}]

%o (PARI) a(n)={(-1)^(n+1) * 2^n * n!^2} \\ _Andrew Howroyd_, Nov 07 2019

%Y Cf. A000079, A001044, A019669, A334381, A334383.

%Y Row of A340591 (in absolute values).

%K sign

%O 0,2

%A _Eric W. Weisstein_

%E Terms a(14) and beyond from _Andrew Howroyd_, Nov 07 2019