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A000407 (2n+1)!/n!.
(Formerly M4270 N1784)
19
1, 6, 60, 840, 15120, 332640, 8648640, 259459200, 8821612800, 335221286400, 14079294028800, 647647525324800, 32382376266240000, 1748648318376960000, 101421602465863680000, 6288139352883548160000, 415017197290314178560000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n>1, a(n)=(1/2)*A001813(n+1). - Zerinvary Lajos, Jun 06 2007

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]

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 in Math. Annalen, 191 (1971), 87-98.

Jolley, Summation of Series, Dover (1961).

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.

Lee A. Newberg, The Number of Clone Orderings, Discrete Applied Mathematics, Vol. 69 (1996), pp. 233-245.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.

H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.

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

P. 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

Dan Levy, Lior Pachter, THE NEIGHBOR-NET ALGORITHM, arXiv:math/0702515v2.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Index to divisibility sequences

Index entries for sequences related to factorial numbers

FORMULA

E.g.f.: (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(4*j+2, j=0..n), (one half of 4-factorial numbers) [ Wolfdieter Lang]

a(n)=C(n+1)*(n+2)!/2; - Paul Barry, Feb 16 2005

For asymptotics see the Robinson paper.

Sum_{n=0..infinity} 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

EXAMPLE

1 + 6*x + 60*x^2 + 840*x^3 + 15120*x^4 + 332640*x^5 + 8648640*x^6 + ...

MAPLE

For Maple program see A000903.

a := n -> pochhammer(n+1, n+1); (for n>=0) [From Peter Luschny, Feb 14 2009]

MATHEMATICA

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, 0, Pochhammer[ n+1, n+1]] (* Michael Somos, May 12 2012 *)

PROG

(PARI) a(n)=(2*n+1)!/n! \\ Charles R Greathouse IV, Jan 12 2012

(Maxima) A000407(n):=(2*n+1)!/n!$

makelist(A000407(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

CROSSREFS

Cf. A001761-A001763, A007696.

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.

Sequence in context: A168478 A101470 A066151 * A099708 A177191 A010040

Adjacent sequences:  A000404 A000405 A000406 * A000408 A000409 A000410

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 29 08:09 EDT 2014. Contains 246187 sequences.