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%I #62 Nov 30 2021 11:38:23
%S 1,4,36,272,2150,16992,134848,1072192,8536914,68036600,542607560,
%T 4329671040,34561892560,275979195520,2204266118400,17609217372416,
%U 140698273234634,1124340854572296,8985828520591912,71822662173752800
%N Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1).
%D I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 2008, p. 456 (Ch. 14, Corollary 2.9).
%H Arthur Cayley, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.hxjnj3&view=1up&seq=211">On the theory of linear transformations</a>, The Cambridge Mathematical Journal, Vol. IV, No. XXIII, February 1845, pp. 193-209. [Accessible only in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H Arthur Cayley, <a href="https://quod.lib.umich.edu/u/umhistmath/ABS3153.0001.001">On the theory of linear transformations</a>, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 80-94. [Accessible through the <a href="https://quod.lib.umich.edu/u/umhistmath/">University of Michigan Historical Math Collection</a>; click on pp. 80 through 94.]
%H Arthur Cayley, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.a0002837516&view=1up&seq=114">On linear transformations</a>, Cambridge and Dublin Mathematical Journal, Vol. I, 1846, pp. 104-122. [Accessible only in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H Arthur Cayley, <a href="https://quod.lib.umich.edu/u/umhistmath/ABS3153.0001.001">On linear transformations</a>, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 95-112. [Accessible through the <a href="https://quod.lib.umich.edu/u/umhistmath/">University of Michigan Historical Math Collection</a>; click on pp. 95 through 112.]
%H I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, <a href="https://doi.org/10.1016/0001-8708(92)90056-Q">Hyperdeterminants</a>, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.9 (p. 246).
%H David G. Glynn, <a href="https://doi.org/10.1017/S0004972700031890">The modular counterparts of Cayley's hyperdeterminants</a>, Bulletin of the Australian Mathematical Society 57(3) (1998), 479-492.
%H Giorgio Ottaviani, Luca Sodomaco, and Emuanuele Ventura, <a href="https://arxiv.org/abs/2008.11670">Asymptotics of degrees and ED degrees of Segre products</a>, arXiv:2008.11670 [math.AG], 2020.
%H Ludwig Schläfli, <a href="http://opacplus.bsb-muenchen.de/title/BV020984320/ft/bsb10942369?page=5">Über die Resultante eines Systemes mehrerer algebraischen Gleichungen, ein Beitrag zur Theorie der Elimination</a>, Denkschr. der Kaiserlicher Akad. der Wiss. math-naturwiss. Klasse, 4 Band, 1852.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperdeterminant.html">Hyperdeterminant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperdeterminant">Hyperdeterminants</a>.
%F a(n) = Sum_{j = 0..n/2} ( (j+n+1)! * 2^(n-2j) )/((j!)^3 * (n-2j)!).
%F a(n) = (n+1)^2*(8*A000172(n)-A000172(n+1))/6. - _Mark van Hoeij_, Jul 02 2010
%F 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
%F a(n) ~ 8^(n+1) / (Pi * 3^(3/2)). - _Vaclav Kotesovec_, Sep 12 2019
%F 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
%e 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]
%p 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);
%p # Second program:
%p a := proc(n) option remember; if n = 0 then return 1 elif n = 1 then return 4 fi;
%p (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:
%p seq(a(n), n=0..19); # _Peter Luschny_, Sep 12 2019
%t 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 *)
%Y Cf. A045899, A050269, A086302, A087981.
%K easy,nonn
%O 0,2
%A _Benjamin J. Young_, Apr 08 2010
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