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

-1, 2, -16, 288, -9216, 460800, -33177600, 3251404800, -416179814400, 67421129932800, -13484225986560000, 3263182688747520000, -939796614359285760000, 317651255653438586880000, -124519292216147926056960000, 56033681497266566725632000000

Offset: 0

Author

Eric W. Weisstein

Keywords

• sign

Comments

Coefficient of the Cayley-Menger determinant of order n.

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

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

|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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Paul C. Kainen, Construction numbers: How to build a graph?, arXiv:2302.13186 [math.CO], 2023.
• Usman A. Khan, Soummya Kar and Jose M. F. Moura, A novel geometric approach towards a linear theory for sensor localization, 2013.
• Alan L. Mackay, On the regular heptagon, J. Math. Chemistry, vol. 21, 1997, 197-209.
• Eric Weisstein's World of Mathematics, Cayley-Menger Determinant.

Crossrefs

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

Row of A340591 (in absolute values).

Programs

Mathematica

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

PARI

a(n)={(-1)^(n+1) * 2^n * n!^2} \\ Andrew Howroyd, Nov 07 2019

Formula

E.g.f.: -arcsinh(x/sqrt(2))^2. - Vladeta Jovovic, Aug 30 2004

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

a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - Terry D. Grant, May 21 2017

From Amiram Eldar, Nov 18 2020: (Start)

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

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

Extensions

Terms a(14) and beyond from Andrew Howroyd, Nov 07 2019