%I #130 Nov 17 2023 11:34:01
%S 1,1,4,144,82944,1194393600,619173642240000,15728001190723584000000,
%T 25569049282962188245401600000000,
%U 3366980847587422591723894776791040000000000,44337041641882947649156022595410930014617600000000000000
%N a(n) = Product_{i=0..n} i!^2.
%C a(n) is the discriminant of the polynomial x(x+1)(x+2)...(x+n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003
%C This is the Hankel transform (see A001906 for definition) of the sequence: 1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, ... (see A000364: Euler numbers). - _Philippe Deléham_, Apr 06 2005
%C Also, for n>0, the quotient of (-1)^(n-1)S(u)^(n^2)/S(un) and the determinant of the (n-1) X (n-1) square matrix [P'(u), P''(u), ..., P^(n-1)(u); P''(u), P'''(u), ..., P^(n)(u); P'''(u), P^(4)(u), ..., P^(n+1)(u); ...; P^(n-1)(u), P^(n)(u), ..., P^(2n-3)(u)] where S and P are the Weierstrass Sigma and The Weierstrass P-function, respectively and f^(n) is the n-th derivative of f. See the King and Schwarz & Weierstrass references. - _Balarka Sen_, Jul 31 2013
%C a(n) is the number of idempotent monotonic labeled magmas. That is, prod(i,j) >= max(i,j) and prod(i,i) = i. - _Chad Brewbaker_, Nov 03 2013
%C Ramanujan's infinite nested radical sqrt(1+2*sqrt(1+3*sqrt(1+...))) = 3 can be written sqrt(1+sqrt(4+sqrt(144+...))) = sqrt(a(1)+sqrt(a(2)+sqrt(a(3)+...))). Vijayaraghavan used that to prove convergence of Ramanujan's formula. - Petros Hadjicostas and _Jonathan Sondow_, Mar 22 2014
%C a(n) is the determinant of the (n+1)-th order Hankel matrix whose (i,j)-entry is equal to A000142(i+j), i,j = 0,1,...,n. - _Michael Shmoish_, Sep 02 2020
%D R. Bruce King, Beyond The Quartic Equation, Birkhauser Boston, Berlin, 1996, p. 72.
%D Srinivasa Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.
%D T. Vijayaraghavan, in Collected Papers of Srinivasa Ramanujan, G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, eds., Cambridge Univ. Press, 1927, p. 348; reprinted by Chelsea, 1962.
%H G. C. Greubel, <a href="/A055209/b055209.txt">Table of n, a(n) for n = 0..32</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry3/barry84r2.html">A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
%H Richard Ehrenborg, <a href="http://www.jstor.org/stable/2589352">The Hankel determinant of exponential polynomials</a>, Amer. Math. Monthly, Vol. 107, No. 6 (2000), pp. 557-560.
%H William Q. Erickson and Jan Kretschmann, <a href="https://arxiv.org/abs/2311.07522">The structure and normalized volume of Monge polytopes</a>, arXiv:2311.07522 [math.CO], 2023. See p. 7.
%H John W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H Christian Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
%H H. A. Schwarz and K. Weierstrass, <a href="https://archive.org/stream/formelnundlehrs01weieuoft#page/18/mode/2up">Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen</a>, Springer, Berlin, 1893, p. 19.
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see pp. 305-306.
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = A000178(n)^2. - _Philippe Deléham_, Mar 06 2004
%F a(n) = Product_{i=0..n} i^(2*n - 2*i + 2). - _Charles R Greathouse IV_, Jan 12 2012
%F Asymptotic: a(n) ~ exp(2*zeta'(-1)-3/2*(1+n^2)-3*n)*(2*Pi)^(n+1)*(n+1)^ (n^2+2*n+5/6). - _Peter Luschny_, Jun 23 2012
%F lim_{n->infinity} a(n)^(2^(-(n+1))) = 1. - _Vaclav Kotesovec_, Jun 06 2015
%F Sum_{n>=0} 1/a(n) = A258619. - _Amiram Eldar_, Nov 17 2020
%p seq(mul(mul(j^2,j=1..k), k=0..n), n=0..10); # _Zerinvary Lajos_, Sep 21 2007
%t Table[Product[(i!)^2,{i,n}],{n,0,11}] (* _Harvey P. Dale_, Jul 06 2011 *)
%t Table[BarnesG[n + 2]^2, {n, 0, 11}] (* _Jan Mangaldan_, May 07 2014 *)
%o (PARI) a(n)=prod(i=1,n,i!)^2 \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Sage)
%o def A055209(n) :
%o return prod(factorial(i)^(2) for i in (0..n))
%o [A055209(n) for n in (0..11)] # _Jani Melik_, Jun 06 2015
%o (Magma) [1] cat [(&*[(Factorial(k))^2: k in [1..n]]): n in [1..10]]; // _G. C. Greubel_, Oct 14 2018
%Y Cf. A055209 is the Hankel transform (see A001906 for definition) of A000023, A000142, A000166, A000522, A003701, A010842, A010843, A051295, A052186, A053486, A053487.
%Y Cf. A112302, A258619.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jul 18 2000