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 A055209 a(n) = Product_{i=0..n} i!^2. 22
 1, 1, 4, 144, 82944, 1194393600, 619173642240000, 15728001190723584000000, 25569049282962188245401600000000, 3366980847587422591723894776791040000000000, 44337041641882947649156022595410930014617600000000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the discriminant of the polynomial x(x+1)(x+2)...(x+n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003 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 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 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 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 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 REFERENCES R. Bruce King, Beyond The Quartic Equation, Birkhauser Boston, Berlin, 1996, p. 72. Srinivasa Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226. 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. LINKS G. C. Greubel, Table of n, a(n) for n = 0..32 Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, Vol. 107, No. 6 (2000), pp. 557-560. William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7. John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp. H. A. Schwarz and K. Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, Springer, Berlin, 1893, p. 19. Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see pp. 305-306. Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1. Index entries for sequences related to factorial numbers FORMULA a(n) = A000178(n)^2. - Philippe Deléham, Mar 06 2004 a(n) = Product_{i=0..n} i^(2*n - 2*i + 2). - Charles R Greathouse IV, Jan 12 2012 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 lim_{n->infinity} a(n)^(2^(-(n+1))) = 1. - Vaclav Kotesovec, Jun 06 2015 Sum_{n>=0} 1/a(n) = A258619. - Amiram Eldar, Nov 17 2020 MAPLE seq(mul(mul(j^2, j=1..k), k=0..n), n=0..10); # Zerinvary Lajos, Sep 21 2007 MATHEMATICA Table[Product[(i!)^2, {i, n}], {n, 0, 11}] (* Harvey P. Dale, Jul 06 2011 *) Table[BarnesG[n + 2]^2, {n, 0, 11}] (* Jan Mangaldan, May 07 2014 *) PROG (PARI) a(n)=prod(i=1, n, i!)^2 \\ Charles R Greathouse IV, Jan 12 2012 (Sage) def A055209(n) : return prod(factorial(i)^(2) for i in (0..n)) [A055209(n) for n in (0..11)] # Jani Melik, Jun 06 2015 (Magma) [1] cat [(&*[(Factorial(k))^2: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018 CROSSREFS Cf. A055209 is the Hankel transform (see A001906 for definition) of A000023, A000142, A000166, A000522, A003701, A010842, A010843, A051295, A052186, A053486, A053487. Cf. A112302, A258619. Sequence in context: A186081 A138176 A203424 * A239350 A343697 A030450 Adjacent sequences: A055206 A055207 A055208 * A055210 A055211 A055212 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jul 18 2000 STATUS approved

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Last modified June 22 14:17 EDT 2024. Contains 373587 sequences. (Running on oeis4.)