

A091868


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


5



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
From Andrew Weimholt, Sep 23 2009: (Start)
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)


LINKS

Table of n, a(n) for n=0..9.
E. W. Weisstein, MathWorld: Fibonacci Polynomial


FORMULA

a(n) = (n!)^(n+1) = a(n1) * n^n * n!.
a(n) = A000178(n)*A002109(n), i.e., product of superfactorials and hyperfactorials.  Henry Bottomley, Nov 13 2009
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
seq(mul(mul(j, j=1..n), k=0..n), n=0..8); # Zerinvary Lajos, Sep 21 2007


MATHEMATICA

Table[(n!)^(n+1), {n, 0, 8}] (* Harvey P. Dale, Apr 30 2012 *)


PROG

(MAGMA) [Factorial(n)^(n+1): n in [0..10]]; // Vincenzo Librandi, Nov 25 2015


CROSSREFS

Cf. A036740.
Sequence in context: A095821 A340562 A176113 * A247732 A162090 A017187
Adjacent sequences: A091865 A091866 A091867 * A091869 A091870 A091871


KEYWORD

easy,nonn,nice


AUTHOR

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


EXTENSIONS

Edited by N. J. A. Sloane, Oct 24 2009 at the suggestion of R. J. Mathar
a(9) from Vincenzo Librandi, Nov 25 2015


STATUS

approved



