A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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

• Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
• 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.
• Index to divisibility sequences.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A000165, A001818, A079484, A197036, A334380.

J1: A002474, J2: A002506, J3: A014401.

Programs

Magma

[4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019

Mathematica

Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)

PARI

a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019

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)

2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017

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