login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A176097 Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1). 1
1, 4, 36, 272, 2150, 16992, 134848, 1072192, 8536914, 68036600, 542607560, 4329671040, 34561892560, 275979195520, 2204266118400, 17609217372416, 140698273234634, 1124340854572296, 8985828520591912, 71822662173752800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 2008, p. 456 (Ch. 14, Corollary 2.9).

LINKS

Table of n, a(n) for n=0..19.

Arthur Cayley, On the theory of linear transformations, The Cambridge Mathematical Journal, Vol. IV, No. XXIII, February 1845, pp. 193-209. [Accessible only in the USA through the Hathi Trust Digital Library.]

Arthur Cayley, On the theory of linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 80-94. [Accessible through the University of Michigan Historical Math Collection; click on pp. 80 through 94.]

Arthur Cayley, On linear transformations, Cambridge and Dublin Mathematical Journal, Vol. I, 1846, pp. 104-122. [Accessible only in the USA through the Hathi Trust Digital Library.]

Arthur Cayley, On linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 95-112. [Accessible through the University of Michigan Historical Math Collection; click on pp. 95 through 112.]

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.9 (p. 246).

David G. Glynn, The modular counterparts of Cayley's hyperdeterminants, Bulletin of the Australian Mathematical Society 57(3) (1998), 479-492.

Ludwig Schläfli, Über die Resultante eines Systemes mehrerer algebraischen Gleichungen, ein Beitrag zur Theorie der Elimination, Denkschr. der Kaiserlicher Akad. der Wiss. math-naturwiss. Klasse, 4 Band, 1852.

Eric Weisstein's World of Mathematics, Hyperdeterminant.

Wikipedia, Hyperdeterminants.

FORMULA

a(n) = Sum_{j = 0..n/2} ( (j+n+1)! * 2^(n-2j) )/((j!)^3 * (n-2j)!).

a(n) = (n+1)^2*(8*A000172(n)-A000172(n+1))/6. - Mark van Hoeij, Jul 02 2010

G.f.: hypergeom([-1/3, 1/3],[1],27*x^2/(1-2*x)^3)*(1-2*x)/((x+1)^2*(1-8*x)). - Mark van Hoeij, Apr 11 2014

a(n) ~ 8^(n+1) / (Pi * 3^(3/2)). - Vaclav Kotesovec, Sep 12 2019

a(n) = (a(n-1)*(21*n^3 - 10*n^2 - 9*n + 6) + a(n-2)*(24*n^3 + 16*n^2))/((3*n - 1)*n^2) for n >= 2. - Peter Luschny, Sep 12 2019

EXAMPLE

For k=1, the hyperdeterminant of the matrix (a_ijk) (for 0 <= i,j,k <= 1) is (a_000 * a_111)^2 + (a001 * a110)^2 + (a_010 * a_101)^2 + (a_011 * a_100)^2 -2(a_000 * a_001 * a_110 * a_111 + a_000 * a_010 * a_101 * a_111 + a_000 * a_011 * a_100 * a_111 + a_001 * a_010 * a_101 * a_110 + a_001 * a_011 * a_110 * a_100 + a_010 * a_011 * a_101 * a_100) + 4(a_000 * a_011 * a_101 * a_110 + a_001 * a_010 * a_100 * a_111) (see Gelfand, Kapranov & Zelevinsky, pp. 2 and 448.) [Corrected by Petros Hadjicostas, Sep 12 2019]

MAPLE

a:= k-> add((j+k+1)! /(j!)^3 /(k-2*j)! *2^(k-2*j), j=0..floor(k/2)): seq(a(n), n=0..20);

# Second program:

a := proc(n) option remember; if n = 0 then return 1 elif n = 1 then return 4 fi;

(a(n-1)*(21*n^3-10*n^2-9*n+6)+a(n-2)*(24*n^3+16*n^2))/((3*n-1)*n^2) end:

seq(a(n), n=0..19); # Peter Luschny, Sep 12 2019

MATHEMATICA

Table[Sum[(j + n + 1)!*2^(n - 2*j)/(j!^3*(n - 2*j)!), {j, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 12 2019 *)

CROSSREFS

Cf. A045899, A050269, A086302, A087981.

Sequence in context: A183496 A043024 A144889 * A173429 A172134 A098916

Adjacent sequences:  A176094 A176095 A176096 * A176098 A176099 A176100

KEYWORD

easy,nonn

AUTHOR

Benjamin J. Young, Apr 08 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 11 07:41 EST 2019. Contains 329914 sequences. (Running on oeis4.)