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%I M2040 N0808 #268 Jul 21 2023 15:50:15
%S 1,2,12,120,1680,30240,665280,17297280,518918400,17643225600,
%T 670442572800,28158588057600,1295295050649600,64764752532480000,
%U 3497296636753920000,202843204931727360000,12576278705767096320000,830034394580628357120000,58102407620643984998400000
%N Quadruple factorial numbers: a(n) = (2n)!/n!.
%C Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
%C Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
%C Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
%C Hankel transform is A137565. - _Paul Barry_, Nov 25 2009
%C The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - _Wolfdieter Lang_, Jan 09 2012
%C From _Tom Copeland_, Nov 15 2014: (Start)
%C Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
%C Equals A000407*2 with leading 1 added. (End)
%C a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - _Luis Manuel Rivera Martínez_, Mar 04 2015
%C Self-convolution gives A076729. - _Vladimir Reshetnikov_, Oct 11 2016
%C For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - _Ivan N. Ianakiev_, Feb 28 2017
%C For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - _Manfred Scheucher_, Dec 29 2017
%D D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
%D L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
%D R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A001813/b001813.txt">Table of n, a(n) for n = 0..100</a>
%H Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, <a href="https://doi.org/10.13140/RG.2.2.25384.83201">Hermite and Laguerre Functions: a Unifying Point of View</a>, Università degli Studi di Catania, Sicily, Italy (2019).
%H Murray Bremner and Martin Markl, <a href="https://arxiv.org/abs/1809.08191">Distributive laws between the Three Graces</a>, arXiv:1809.08191 [math.AT], 2018.
%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 Peter J. Cameron, <a href="https://doi.org/10.1093/qmath/38.2.155">Some treelike objects</a>, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Elliot J. Carr and Matthew J. Simpson, <a href="https://arxiv.org/abs/1810.08890">New homogenization approaches for stochastic transport through heterogeneous media</a>, arXiv:1810.08890 [physics.bio-ph], 2018.
%H W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, <a href="https://arxiv.org/abs/math/0503300">Parity reversing involutions on plane trees and 2-Motzkin paths</a>, arXiv:math/0503300 [math.CO], 2005.
%H Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, <a href="https://arxiv.org/abs/2004.04203">On recursively defined combinatorial classes and labelled trees</a>, arXiv:2004.04203 [math.CO], 2020.
%H P. Cvitanovic, <a href="http://www.nbi.dk/~predrag/papers/PRD14-76.pdf">Group theory for Feynman diagrams in non-Abelian gauge theories</a>, Phys. Rev. D14 (1976), 1536-1553.
%H Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.
%H John Engbers, David Galvin, and Clifford Smyth, <a href="https://arxiv.org/abs/1610.05803">Restricted Stirling and Lah numbers and their inverses</a>, arXiv:1610.05803 [math.CO], 2016. See p. 8.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 127
%H S. Goodenough and C. Lavault, <a href="http://arxiv.org/abs/1404.1894">On subsets of Riordan subgroups and Heisenberg--Weyl algebra</a>, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
%H S. Goodenough and C. Lavault, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p16/0">Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups</a>, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
%H H. W. Gould, <a href="/A007680/a007680.pdf">A class of binomial sums and a series transform</a>, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
%H A. M. Ibrahim, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.pdf">Extension of factorial concept to negative numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=115">Encyclopedia of Combinatorial Structures 115</a>
%H L. C. Larson, <a href="/A000900/a000900_1.pdf">The number of essentially different nonattacking rook arrangements</a>, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
%H Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, <a href="http://ajc.maths.uq.edu.au/pdf/52/ajc_v52_p041.pdf">On the number of mth roots of permutations</a>, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
%H Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, <a href="http://arxiv.org/abs/1005.1531">On the number of mth roots of permutations</a>, arXiv:1005.1531 [math.CO], 2010-2011.
%H Édouard Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
%H R. J. Marsh and P. P. Martin, <a href="https://arxiv.org/abs/math/0612572">Pascal arrays: counting Catalan sets</a>, arXiv:math/0612572 [math.CO], 2006.
%H Calin D. Morosan, <a href="http://dx.doi.org/10.1016/j.ipl.2006.06.007">On the number of broadcast schemes in networks</a>, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
%H J. Riordan, <a href="/A001813/a001813.pdf">Letter to N. J. A. Sloane, Feb 03 1975</a> (with notes by njas)
%H H. E. Salzer, <a href="http://dx.doi.org/10.1090/S0025-5718-1948-0026413-5">Coefficients for expressing the first thirty powers in terms of the Hermite polynomials</a>, Math. Comp., 3 (1948), 167-169.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F E.g.f.: (1-4*x)^(-1/2).
%F a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2).
%F Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)), x=0..infinity), n=0, 1, .. . This representation is unique. - _Karol A. Penson_, Sep 18 2001
%F Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - _Benoit Cloitre_, Apr 27 2003
%F With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - _Paul Barry_, May 09 2003
%F a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%F For asymptotics, see the Robinson paper.
%F a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - _André F. Labossière_, Jun 21 2007
%F a(n) = 12*A051618(a) n >= 2. - _Zerinvary Lajos_, Feb 15 2008
%F a(n) = A000984(n)*A000142(n). - _Zerinvary Lajos_, Mar 25 2008
%F a(n) = A016825(n-1)*a(n-1). - _Roger L. Bagula_, Sep 17 2008
%F a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
%F From _Paul Barry_, Jan 15 2009: (Start)
%F G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
%F a(n) = (n+1)!*A000108(n). (End)
%F a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - _Philippe Deléham_, Feb 10 2009
%F G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - _Paul Barry_, Nov 25 2009
%F a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - _Reinhard Zumkeller_, Feb 19 2010
%F From _Gary W. Adamson_, Jul 19 2011: (Start)
%F a(n) = upper left term 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 ...
%F (End)
%F a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012
%F G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 18 2013
%F G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013
%F D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - _Ivan N. Ianakiev_, Aug 07 2013
%F Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - _Ilya Gutkovskiy_, Nov 10 2016
%F Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - _Amiram Eldar_, Feb 20 2021
%e The following permutations of order 8 and their reversals have this property:
%e 1 7 3 5 2 4 0 6
%e 1 7 4 2 5 3 0 6
%e 2 3 7 6 1 0 4 5
%e 2 4 7 1 6 0 3 5
%e 3 2 6 7 0 1 5 4
%e 3 5 1 7 0 6 2 4
%p A001813 := n->(2*n)!/n!;
%p A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])): seq(A001813(n), n=0..16);
%p # _Peter Luschny_, Jun 23 2011
%t Table[(2n)!/n!, {n,0,20}] (* _Harvey P. Dale_, May 02 2011 *)
%o (Sage) [binomial(2*n,n)*factorial(n) for n in range(0, 17)] # _Zerinvary Lajos_, Dec 03 2009
%o (PARI) a(n)=binomial(n+n,n)*n! \\ _Charles R Greathouse IV_, Jun 15 2011
%o (PARI) first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ _Iain Fox_, Jan 01 2018 (corrected by _Iain Fox_, Jan 11 2018)
%o (Maxima) makelist(binomial(n+n, n)*n!,n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [Factorial(2*n)/Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Oct 09 2018
%o (GAP) List([0..20],n->Factorial(2*n)/Factorial(n)); # _Muniru A Asiru_, Nov 01 2018
%o (Python)
%o from math import factorial
%o def A001813(n): return factorial(n<<1)//factorial(n) # _Chai Wah Wu_, Feb 14 2023
%Y Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
%Y Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
%Y Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
%Y Cf. A000085, A135401, A297708. - _Manfred Scheucher_, Jan 07 2018
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 01 2000
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