A176097 Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1).

1, 4, 36, 272, 2150, 16992, 134848, 1072192, 8536914, 68036600, 542607560, 4329671040, 34561892560, 275979195520, 2204266118400, 17609217372416, 140698273234634, 1124340854572296, 8985828520591912, 71822662173752800

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, 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.

Giorgio Ottaviani, Luca Sodomaco, and Emuanuele Ventura, Asymptotics of degrees and ED degrees of Segre products, arXiv:2008.11670 [math.AG], 2020.

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 * A345633 A173429 A172134

Adjacent sequences: A176094 A176095 A176096 * A176098 A176099 A176100

Keyword

easy,nonn

Author

Benjamin J. Young, Apr 08 2010

Status

approved