This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002454 M3693 N1510 #109 Aug 18 2025 23:36:13
%S A002454 1,4,64,2304,147456,14745600,2123366400,416179814400,106542032486400,
%T A002454 34519618525593600,13807847410237440000,6682998146554920960000,
%U A002454 3849406932415634472960000,2602199086312968903720960000,2040124083669367620517232640000,1836111675302430858465509376000000
%N A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.
%C A002454 Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
%C A002454 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
%C A002454 From _Zhi-Wei Sun_, Jun 26 2022: (Start)
%C A002454 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.
%C A002454 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)
%D A002454 Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
%D A002454 Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
%D A002454 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.
%D A002454 E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
%D A002454 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002454 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002454 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002454 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.
%H A002454 T. D. Noe, <a href="/A002454/b002454.txt">Table of n, a(n) for n = 0..50</a>
%H A002454 T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/7/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
%H A002454 Han Wang and Zhi-Wei Sun, <a href="http://arxiv.org/abs/2206.02589">Proof of a conjecture involving derangements and roots of unity</a>, arXiv:2206.02589 [math.CO], 2022.
%H A002454 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H A002454 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A002454 (-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
%F A002454 E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - _Vladimir Kruchinin_, Sep 12 2010
%F A002454 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
%F A002454 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
%F A002454 From _Ilya Gutkovskiy_, Dec 02 2016: (Start)
%F A002454 a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
%F A002454 Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
%F A002454 From _Daniel Suteu_, Dec 02 2016: (Start)
%F A002454 a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
%F A002454 a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
%F A002454 2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - _Daniel Suteu_, Feb 03 2017
%F A002454 Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - _Daniel Suteu_, Nov 01 2017
%F A002454 Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - _Amiram Eldar_, Apr 09 2022
%F A002454 Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - _Daniel Suteu_, Apr 09 2022
%t A002454 Array[4^# (#!)^2 &, 14, 0] (* _Michael De Vlieger_, Nov 01 2017 *)
%o A002454 (PARI) a(n) = 4^n*(n!)^2; \\ _Michel Marcus_, Mar 13 2019
%o A002454 (Magma) [4^n*Factorial(n)^2: n in [0..15]]; // _Vincenzo Librandi_, Mar 15 2019
%Y A002454 Cf. A000165, A001818, A079484, A197036, A334380.
%Y A002454 J1: A002474, J2: A002506, J3: A014401.
%K A002454 nonn,easy,changed
%O A002454 0,2
%A A002454 _N. J. A. Sloane_