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A002454 Central factorial numbers: a(n) = 4^n (n!)^2.
(Formerly M3693 N1510)
10
1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000 (list; graph; refs; listen; history; text; internal format)
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

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).

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..50

Index to divisibility sequences

Index entries for sequences related to factorial numbers

FORMULA

(-1)^n*a(n) is the coefficient of x^1 in prod(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->infinity} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017

MATHEMATICA

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

CROSSREFS

Cf. A000165, A001818.

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

Sequence in context: A081559 A298433 A261042 * A013043 A296741 A167406

Adjacent sequences:  A002451 A002452 A002453 * A002455 A002456 A002457

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 15 09:12 EDT 2018. Contains 313756 sequences. (Running on oeis4.)