|
| |
|
|
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 (zerinvarylajos(AT)yahoo.com), 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]
|
|
|
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.
DAN LEVY AND LIOR PACHTER, THE NEIGHBOR-NET ALGORITHM, arXiv:math/0702515v2,
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
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
|
|
|
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
|
| |
|
|
