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A000407
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a(n) = (2*n+1)! / n!.
(Formerly M4270 N1784)
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45
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1, 6, 60, 840, 15120, 332640, 8648640, 259459200, 8821612800, 335221286400, 14079294028800, 647647525324800, 32382376266240000, 1748648318376960000, 101421602465863680000, 6288139352883548160000, 415017197290314178560000
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OFFSET
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0,2
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COMMENTS
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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
The product of the first 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
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REFERENCES
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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).
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LINKS
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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.
Peter J. Cameron, Sequences Realized by Oligomorphic Permutation Groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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.
Index to divisibility sequences
Index entries for sequences related to factorial numbers
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FORMULA
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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.
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EXAMPLE
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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
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MAPLE
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For Maple program see A000903.
a := n -> pochhammer(n+1, n+1); (for n>=0) # Peter Luschny, Feb 14 2009
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MATHEMATICA
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Table[(2n + 1)!/n!, {n, 0, 30}] (* Stefan Steinerberger, Apr 08 2006 *)
s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 5, 5!, 4}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
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 *)
a[n_] := Range[n + 1, 2n + 1]; b[n_] := Times@@a[n]; Table[b[n], {n, 0, 17}] (* Carlos Eduardo Olivieri, Jun 11 2015 *)
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PROG
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(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
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CROSSREFS
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Cf. A001761-A001763, A007696, A024199, A073010.
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).
Column m=0 of A292219.
Sequence in context: A101470 A066151 A339191 * A099708 A177191 A010040
Adjacent sequences: A000404 A000405 A000406 * A000408 A000409 A000410
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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