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%I #112 Sep 08 2022 08:44:56
%S 1,4,32,384,6144,122880,2949120,82575360,2642411520,95126814720,
%T 3805072588800,167423193907200,8036313307545600,417888291992371200,
%U 23401744351572787200,1404104661094367232000,89862698310039502848000
%N a(n) = 4^n * n!.
%C Original name was "Quadruple factorial numbers".
%C For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_4)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
%C Number of n X n monomial matrices with entries 0, +/-1, +/-i.
%C a(n) is the product of the positive integers <= 4*n that are multiples of 4. - _Peter Luschny_, Jun 23 2011
%C Also, a(n) is the number of signed permutations of length 2*n that are equal to their reverse-complements. (See the Hardt and Troyka reference.) - _Justin M. Troyka_, Aug 13 2011.
%C Pi^n/a(n) is the volume of a 2*n-dimensional ball with radius 1/2. - _Peter Luschny_, Jul 24 2012
%C Equals the first right hand column of A167557, and also equals the first right hand column of A167569. - _Johannes W. Meijer_, Nov 12 2009
%C a(n) is the order of the group U_n(Z[i]) = {A in M_n(Z[i]): A*A^H = I_n}, the group of n X n unitary matrices over the Gaussian integers. Here A^H is the conjugate transpose of A. - _Jianing Song_, Mar 29 2021
%H Vincenzo Librandi, <a href="/A047053/b047053.txt">Table of n, a(n) for n = 0..100</a>
%H R. B. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalizing Tuenter's Binomial Sums</a>, J. Int. Seq. 18 (2015), # 15.3.2.
%H CombOS - Combinatorial Object Server, <a href="http://combos.org/cperm">Generate colored permutations</a>
%H R. Coquereaux and J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), <a href="https://doi.org/10.1142/S0218216516500474">DOI</a>.
%H Sylvie Corteel and Lauren Williams, <a href="http://arxiv.org/abs/0810.2916">Tableaux Combinatorics for the Asymmetric Exclusion Process II</a>, arXiv:0810.2916 [math.CO], 2008-2009. [From _Jonathan Vos Post_, Oct 17 2008]
%H A. Hardt and J. M. Troyka, <a href="http://www.mat.unisi.it/newsito/puma/public_html/23_3/hardt_troyka.pdf">Restricted symmetric signed permutations</a>, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179-217.
%H A. Hardt and J. M. Troyka, <a href="https://apps.carleton.edu/curricular/math/assets/Andy_hardt_slides.pdf">Slides</a> (associated with the Hardt and Troyka reference above).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=492">Encyclopedia of Combinatorial Structures 492</a>.
%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, 5 (2002), Article 02.1.7.
%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications</a>, J. Int. Seq. 13 (2010), Article 10.6.7, p. 39.
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, 9 (2006), Article 06.1.1.
%F a(n) = 4^n * n!.
%F E.g.f.: 1/(1 - 4*x).
%F Integral representation as the n-th moment of a positive function on a positive half-axis: in Maple notation a(n) = int(x^n*exp(-4*x)/4, x=0..infinity), n = 0, 1, ... This representation is unique. - _Karol A. Penson_, Jan 28 2002
%F Sum_{k>=0} (-1)^k/(2*k + 1)^n = (-1)^n * n * (PolyGamma[n-1, 1/4] - PolyGamma[n-1, 3/4]) / a(n) for n > 0. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 27 2006
%F a(n) = Sum_{k=0..n} C(n,k)*(2k)!*(2(n-k))!/(k!(n-k)!) = Sum_{k=0..n} C(n,k)*A001813(k)*A001813(n-k). - _Paul Barry_, May 04 2007
%F E.g.f.: With interpolated zeros, 1 + sqrt(pi)*x*exp(x^2)*erf(x). - _Paul Barry_, Apr 10 2010
%F From _Gary W. Adamson_, Jul 19 2011: (Start)
%F a(n) = sum of top row terms of M^n, M = an infinite square production matrix as follows:
%F 2, 2, 0, 0, 0, 0, ...
%F 4, 4, 4, 0, 0, 0, ...
%F 6, 6, 6, 6, 0, 0, ...
%F 8, 8, 8, 8, 8, 0, ...
%F ... (End)
%F G.f.: 1/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 8*x/(1 - 12*x/(1 - 12*x/(1 - 16*x/1 - ... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k + 1)/(8*x*(k + 1) - 1 + 8*x*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(2*k + 1) - 16*x^2*(k + 1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 28 2013
%F a(n) = A000142(n) * A000302(n). - _Michel Marcus_, Nov 28 2013
%F a(n) = A087299(2*n). - _Michael Somos_, Jan 03 2015
%F D-finite with recurrence: a(n) - 4*n*a(n-1) = 0. - _R. J. Mathar_, Jan 27 2020
%F From _Amiram Eldar_, Jun 25 2020: (Start)
%F Sum_{n>=0) 1/a(n) = e^(1/4) (A092042).
%F Sum_{n>=0) (-1)^n/a(n) = e^(-1/4) (A092616). (End)
%e G.f. = 1 + 4*x + 32*x^2 + 384*x^3 + 6144*x^4 + 122880*x^5 + 2949120*x^6 + ...
%p A047053:= n -> mul(k, k = select(k-> k mod 4 = 0, [$1..4*n])): seq(A047053(n), n = 0.. 16); # _Peter Luschny_, Jun 23 2011
%t a[n_]:= With[{m=2n}, If[ m<0, 0, m!*SeriesCoefficient[1 +Sqrt[Pi]*x*Exp[x^2]*Erf[x], {x, 0, m}]]]; (* _Michael Somos_, Jan 03 2015 *)
%t Table[4^n n!,{n,0,20}] (* _Harvey P. Dale_, Sep 19 2021 *)
%o (PARI) a(n)=4^n*n!;
%o (Magma) [4^n*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 20 2011
%Y Cf. A000142, A000165, A007696, A008545, A032031, A047058, A087299, A092042, A092616.
%Y a(n)= A051142(n+1, 0) (first column of triangle).
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
%E Edited by _Karol A. Penson_, Jan 22 2002
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