OFFSET
0,2
COMMENTS
Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)
REFERENCES
Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
FORMULA
(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022
MATHEMATICA
Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)
PROG
(PARI) a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019
(Magma) [4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved