

A091868


a(n) = (n!)^(n+1).


9



1, 1, 8, 1296, 7962624, 2985984000000, 100306130042880000000, 416336312719673760153600000000, 281633758444745849464726940024832000000000, 39594086612242519324387557078266845776303882240000000000
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OFFSET

0,3


COMMENTS

Let f(x) be a monic polynomial of degree n. Let u be any number and let m be the matrix of values f(u+ij) for i,j=1..n. Then the determinant of m is a(n).  T. D. Noe, Aug 23 2008
Also, number of ways to assemble an nsimplex from n+1 labeled (n1)simplices with labeled vertices, where lefthanded and righthanded counterparts are considered equivalent.
For n=2, we are constructing a triangle from 3 labeled linesegments with labeled endpoints. Solutions which differ by a rotation or a reflection are considered equivalent. Because reflections are equivalent, there is only 1 way to order the linesegments, and each linesegment can be oriented in 2 ways, so the total number of solutions is 2^3 = 8. For n=3, we are constructing a tetrahedron from 4 labeled triangles with labeled vertices. Without loss of generality, we can pick one labeled triangle to serve as our face of reference. For this face, we do not care which side of the triangle will face the interior of the tetrahedron as this just translates into a reflection of the tetrahedron, nor do we care about which rotation we pick as these just translate into rotations of the tetrahedron. From this reference triangle, there are 6 (=3!) ways to assign the remaining triangles to the faces of the tetrahedron, and each triangle can be oriented in 6 (=3!) ways (we can pick which side of the triangle will face the interior of the tetrahedron, and we can pick from 3 rotations). This gives 6^4 solutions.
Cf. A165644 (same idea, but reflections are distinct). A165642 and A165643 are the corresponding sequences for cubes instead of simplices. (End)
a(n) is the number of preference profiles in the stable marriage problem with n women and n men, where all the men rank women in the same order. Given such a profile, the GaleShapley menproposing algorithm ends in n rounds. Equivalently, this is the number of preference profiles where all the women rank men in the same order.  Tanya Khovanova and MIT PRIMES STEP Senior group, May 23 2021
a(n1) is the determinant of the n X n matrix with elements m(i,j) = s(n+i1,j), 1 <= i <= n, 1 <= j <= n, where s(x,y) are the unsigned Stirling numbers of the first kind.  Fabio Visonà, May 22 2022


LINKS

Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.


FORMULA

a(n) = (n!)^(n+1) = a(n1) * n^n * n!.
a(n) ~ (2*Pi)^((n+1)/2) * n^((n+1)*(2*n+1)/2) / exp(n^2 + n  1/12).  Vaclav Kotesovec, Jul 10 2015


MAPLE

(n!)^(n+1);
a[0]:=1:for n from 1 to 20 do a[n]:=product(n!, k=0..n) od: seq(a[n], n=0..8); # Zerinvary Lajos, Jun 11 2007


MATHEMATICA



PROG



CROSSREFS



KEYWORD

easy,nonn,nice


AUTHOR

Nicolau C. Saldanha (nicolau(AT)mat.pucrio.br), Mar 10 2004


EXTENSIONS



STATUS

approved



