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%I M4270 N1784 #169 Feb 20 2024 02:20:38
%S 1,6,60,840,15120,332640,8648640,259459200,8821612800,335221286400,
%T 14079294028800,647647525324800,32382376266240000,1748648318376960000,
%U 101421602465863680000,6288139352883548160000,415017197290314178560000
%N a(n) = (2*n+1)! / n!.
%C The e.g.f. of 1/a(n) = n!/(2*n+1)! is (exp(sqrt(x)) - exp(-sqrt(x)))/(2*sqrt(x)). - _Wolfdieter Lang_, Jan 09 2012
%C Product of the larger parts of the partitions of 2n+2 into exactly two parts. - _Wesley Ivan Hurt_, Jun 15 2013
%C For n > 0, a(n-1) = (2n-1)!/(n-1)!, the number of ways n people can line up in n labeled queues. The derivation is straightforward. Person 1 has (2n-1) choices - be first in line in one of the queues or get behind one of the other people. Person 2 has (2n-2) choices - choose one of the n queues or get behind one of the remaining n-2 people. Continuing in this fashion, we finally find that person n has to choose one of the n queues. - _Dennis P. Walsh_, Mar 24 2016
%C For n > 0, a(n-1) is the number of functions f:[n]->[2n] that are acyclic and injective. Note that f is acyclic if, for all x in [n], x is not a member of the set {f(x),f(f(x)), f(f(f(x))), ...}. - _Dennis P. Walsh_, Mar 25 2016
%C a(n) is the number of labeled maximal outerplanar graphs with n-3 vertices. - _Allan Bickle_, Feb 19 2024
%D L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted with a slightly different title in Math. Annalen, 191 (1971), 87-98.
%D L. B. W. Jolley, Summation of Series, Dover, 1961.
%D Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%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="/A000407/b000407.txt">Table of n, a(n) for n = 0..100</a>
%H L. W. Beineke and R. E. Pippert, <a href="https://doi.org/10.1007/BF02330563">Enumerating labeled k-dimensional trees and ball dissections</a>, Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970, pp. 12-26. Reprinted with a slightly different title in Math. Annalen, Vol. 191 (1971), pp. 87-98.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Peter J. Cameron, <a href="https://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 G. Hu, <a href="https://ieeexplore.ieee.org/document/6083260">Catalan number and enumeration of maximal outerplanar graphs</a>, Tsinghua Science and Technology 25 5 1 (2000), 109-114.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=139">Encyclopedia of Combinatorial Structures 139</a>.
%H Loren 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 Dan Levy and Lior Pachter, <a href="https://arxiv.org/abs/math/0702515">The Neighbor-Net Algorithm</a>, arXiv:math/0702515 [math.CO], 2007-2008.
%H Lee A. Newberg, <a href="https://doi.org/10.1016/0166-218X(96)00093-5">The Number of Clone Orderings</a>, Discrete Applied Mathematics, Vol. 69, No. 3 (1996), pp. 233-245.
%H J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H Robert W. Robinson, <a href="https://doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%H Herbert E. Salzer, <a href="https://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., Vol. 3, No. 23 (1948), pp. 167-169.
%H Herbert E. Salzer, <a href="https://doi.org/10.1090/S0025-5718-1955-0078498-1">Orthogonal polynomials arising in the evaluation of inverse Laplace transforms</a>, Math. Comp. Vol. 9, No. 52 (1955), pp. 164-177.
%H Herbert E. Salzer, <a href="/A000407/a000407.pdf">Orthogonal polynomials arising in the evaluation of inverse Laplace transforms</a>, Math. Comp., Vol. 9, No. 52 (1955), 164-177. [Annotated scanned copy]
%H Maxie 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., Vol. 13 (2010), Article 10.6.7, page 39.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/William_Brouncker,_2nd_Viscount_Brouncker">William Brouncker, 2nd Viscount Brouncker</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F E.g.f.: (1 - 4*x)^(-3/2). - _Michael Somos_, Jan 03 2015
%F E.g.f.: Sum_{k>=0} a(k+2) * x^k / k! = (1 - 2*x - sqrt(1 - 4*x)) / 4.
%F E.g.f. for a(n-1), n >= 0, with a(-1) := 0 is (-1+1/(1-4*x)^(1/2))/2. 2*a(n) = (4*n+2)(!^4) := Product_{j=0..n} (4*j + 2), (one half of 4-factorial numbers). - _Wolfdieter Lang_
%F a(n) = C(n+1)*(n+2)!/2 for all n>=0. - _Paul Barry_, Feb 16 2005
%F For n>1, a(n) = (1/2)*A001813(n+1). - _Zerinvary Lajos_, Jun 06 2007
%F For asymptotics see the Robinson paper.
%F Sum_{n >=0} n!/a(n) = 2*Pi/3^(3/2) = 1.2091995761... [Jolley eq 261]
%F G.f.: 1 / (1 - 6*x / (1 - 4*x / (1 - 10*x / (1 - 8*x / (1 - 14*x / ... ))))). - _Michael Somos_, May 12 2012
%F G.f.: 1/Q(0), where Q(k) = 1 + 2*(2*k-1)*x - 4*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+3)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 02 2013
%F a(n) = -(-1)^n / (4 * a(-2-n)) = a(n-1) * (4*n+2) for all n in Z. - _Michael Somos_, Jan 03 2015
%F a(n) = A087299(2*n + 1). - _Michael Somos_, Jan 03 2015
%F From _Peter Bala_, Feb 16 2015: (Start)
%F Recurrence equation: a(n) = 4*a(n-1) + 4*(2*n - 1)^2*a(n-2) with a(0) = 1 and a(1) = 6.
%F The integer sequence b(n) := a(n)*Sum_{k = 0..n} (-1)^k/(2*k + 1), beginning [1, 4, 52, 608, 12624, ...], satisfies the same second-order recurrence equation. This leads to Brouncker's generalized continued fraction expansion Sum_{k >= 0} (-1)^k/(2*k + 1) = Pi/4 = 1/(1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... )))). Note b(n) = 2^n*A024199(n+1).
%F Recurrence equation: a(n) = (5*n + 2)*a(n-1) - 2*n*(2*n - 1)^2*a(n-2) with a(0) = 1 and a(1) = 6.
%F The integer sequence c(n) := a(n)*Sum_{k = 0..n} k!^2/(2*k + 1)!, beginning [1, 7, 72, 1014, 18276, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion Sum_{k >= 0} k!^2/(2*k + 1)! = 2*Pi/sqrt(27) = 2*A073010 = 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). (End)
%F a(n) = Product_{k=n+1..(2*n+1)} k. - _Carlos Eduardo Olivieri_, Jun 03 2015
%F From _Ilya Gutkovskiy_, Jan 17 2017: (Start)
%F a(n) ~ 2^(2*n+3/2)*n^(n+1)/exp(n).
%F Sum_{n>=0} 1/a(n) = exp(1/4)*sqrt(Pi)*erf(1/2) = 1.184593072938653151..., where erf() is the error function. (End)
%F Sum_{n>=0} (-1)^n/a(n) = exp(-1/4)*sqrt(Pi)*erfi(1/2), where erfi() is the imaginary error function. - _Amiram Eldar_, Jan 18 2021
%F It follows from the comments above that we have a(n) = a(n-1)*(4*n+2), with a(1) = 6, a(0) = 1.
%e G.f. = 1 + 6*x + 60*x^2 + 840*x^3 + 15120*x^4 + 332640*x^5 + 8648640*x^6 + ...
%e For n=1 the a(1)=6 ways for 2 people to line up in 2 queues are as follows: Q1<P1,P2> Q2<>, Q1<P2,P1> Q2<>, Q1<P1> Q2<P2>, Q1<P2> Q2<P1>, Q1<> Q2<P1,P2>, Q1<> Q2<P2,P1>. - _Dennis P. Walsh_, Mar 24 2016
%e For the unique maximal outerplanar graph with 4 vertices, there are C(4,2)=6 ways to label the two degree 3 vertices, and the other two labels are forced. Thus a(1) = 6.
%p For Maple program see A000903.
%p a := n -> pochhammer(n+1,n+1); (for n>=0) # _Peter Luschny_, Feb 14 2009
%t Table[(2n + 1)!/n!, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 08 2006 *)
%t a[ n_] := If[ n < 0, 1/2, 1] Pochhammer[ n + 1, n + 1]; (* _Michael Somos_, Jan 03 2015 *)
%t a[ n_] := Which[ n < -1, -(-1)^n / (4 a[-n - 2]), n == -1, 1/2, True, (2 n + 1)! / n!]; (* _Michael Somos_, Jan 03 2015 *)
%o (PARI) a(n)=(2*n+1)!/n! \\ _Charles R Greathouse IV_, Jan 12 2012
%o (PARI) {a(n) = if( n<-1, -(-1)^n / (4 * a(-n-2)), n==-1, 1/2, (2*n + 1)! / n!))}; /* _Michael Somos_, Jan 03 2015 */
%o (Maxima) A000407(n):=(2*n+1)!/n!$
%o makelist(A000407(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [Factorial(2*n+1) / Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Jun 16 2015
%Y Cf. A001761-A001763, A007696, A024199, A073010.
%Y A100622 is the "Number of topologically distinct solutions to the clone ordering problem for n clones" without the restriction that they be in a single contig (see [Newberg] for definition of contig).
%Y Column m=0 of A292219.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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