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A055209 a(n) = Product_{i=0..n} i!^2. 20
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

REFERENCES

R. B. King, Beyond The Quartic Equation, Birkhauser Boston, Berlin, 1996, p. 72.

S. Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.

H. A. Schwarz & K. Weierstrass, Formeln und Lehrsatzee zum Gebrauche der Elliptischen Functionen, Springer, Berlin, 1893, p. 19.

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

Table of n, a(n) for n=0..10.

P. 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.

R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 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^(2n - 2i + 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) = sqrt(1*sqrt(2*sqrt(3*...))) = Somos's quadratic recurrence constant A112302. - Petros Hadjicostas and Jonathan Sondow, Mar 22 2014 [This formula is incorrect, limit is equal to 1. - Vaclav Kotesovec, Jun 06 2015]

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

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.

Sequence in context: A186081 A138176 A203424 * A239350 A030450 A041629

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 September 22 00:25 EDT 2017. Contains 292326 sequences.