OFFSET
0,2
COMMENTS
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
Product of the larger parts of the partitions of 2n+2 into exactly two parts. - Wesley Ivan Hurt, Jun 15 2013
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
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
a(n) is the number of labeled maximal outerplanar graphs with n-3 vertices. - Allan Bickle, Feb 19 2024
REFERENCES
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.
L. B. W. Jolley, Summation of Series, Dover, 1961.
Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
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).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..100
L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, 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.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Peter J. Cameron, Sequences Realized by Oligomorphic Permutation Groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Hu, Catalan number and enumeration of maximal outerplanar graphs, Tsinghua Science and Technology 25 5 1 (2000), 109-114.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 139.
Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Dan Levy and Lior Pachter, The Neighbor-Net Algorithm, arXiv:math/0702515 [math.CO], 2007-2008.
Lee A. Newberg, The Number of Clone Orderings, Discrete Applied Mathematics, Vol. 69, No. 3 (1996), pp. 233-245.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Robert W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
Herbert E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., Vol. 3, No. 23 (1948), pp. 167-169.
Herbert E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. Vol. 9, No. 52 (1955), pp. 164-177.
Herbert E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp., Vol. 9, No. 52 (1955), 164-177. [Annotated scanned copy]
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq., Vol. 13 (2010), Article 10.6.7, page 39.
Wikipedia, William Brouncker, 2nd Viscount Brouncker.
FORMULA
E.g.f.: (1 - 4*x)^(-3/2). - Michael Somos, Jan 03 2015
E.g.f.: Sum_{k>=0} a(k+2) * x^k / k! = (1 - 2*x - sqrt(1 - 4*x)) / 4.
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
a(n) = C(n+1)*(n+2)!/2 for all n>=0. - Paul Barry, Feb 16 2005
For n>1, a(n) = (1/2)*A001813(n+1). - Zerinvary Lajos, Jun 06 2007
For asymptotics see the Robinson paper.
Sum_{n >=0} n!/a(n) = 2*Pi/3^(3/2) = 1.2091995761... [Jolley eq 261]
G.f.: 1 / (1 - 6*x / (1 - 4*x / (1 - 10*x / (1 - 8*x / (1 - 14*x / ... ))))). - Michael Somos, May 12 2012
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
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
a(n) = -(-1)^n / (4 * a(-2-n)) = a(n-1) * (4*n+2) for all n in Z. - Michael Somos, Jan 03 2015
a(n) = A087299(2*n + 1). - Michael Somos, Jan 03 2015
From Peter Bala, Feb 16 2015: (Start)
Recurrence equation: a(n) = 4*a(n-1) + 4*(2*n - 1)^2*a(n-2) with a(0) = 1 and a(1) = 6.
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).
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.
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)
a(n) = Product_{k=n+1..(2*n+1)} k. - Carlos Eduardo Olivieri, Jun 03 2015
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 2^(2*n+3/2)*n^(n+1)/exp(n).
Sum_{n>=0} 1/a(n) = exp(1/4)*sqrt(Pi)*erf(1/2) = 1.184593072938653151..., where erf() is the error function. (End)
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
It follows from the comments above that we have a(n) = a(n-1)*(4*n+2), with a(1) = 6, a(0) = 1.
EXAMPLE
G.f. = 1 + 6*x + 60*x^2 + 840*x^3 + 15120*x^4 + 332640*x^5 + 8648640*x^6 + ...
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
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.
MAPLE
For Maple program see A000903.
a := n -> pochhammer(n+1, n+1); (for n>=0) # Peter Luschny, Feb 14 2009
MATHEMATICA
Table[(2n + 1)!/n!, {n, 0, 30}] (* Stefan Steinerberger, Apr 08 2006 *)
a[ n_] := If[ n < 0, 1/2, 1] Pochhammer[ n + 1, n + 1]; (* Michael Somos, Jan 03 2015 *)
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 *)
PROG
(PARI) a(n)=(2*n+1)!/n! \\ Charles R Greathouse IV, Jan 12 2012
(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 */
(Maxima) A000407(n):=(2*n+1)!/n!$
makelist(A000407(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(Magma) [Factorial(2*n+1) / Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jun 16 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved