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A000680
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(2n)!/2^n.
(Formerly M4287 N1793)
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32
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1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 2375880867360000, 548828480360160000, 151476660579404160000, 49229914688306352000000, 18608907752179801056000000, 8094874872198213459360000000, 4015057936610313875842560000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Denominators in the expansion of cos(sqrt(2)*x) = 1 - (sqrt(2)*x)^2/2! + (sqrt(2)*x)^4/4! - (sqrt(2)*x)^6/6! + ... = 1 - x^2 + x^4/6 - x^6/90 + ... By Stirling's formula in A000142: a(n) ~ 2^(n+1) * (n/e)^(2n) * sqrt(Pi*n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
a(n) is also the constant term in the product : product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^2. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 12 2002
a(n) is also the number of lattice paths in the n-dimensional lattice [0..2]^n. - T. D. Noe (noe(AT)sspectra.com), Jun 06 2002
Representation as the n-th moment of a positive function on the positive half-axis: in Maple notation a(n)=int(x^n*exp(-sqrt(2*x))/sqrt(2*x), x=0..infinity),n=0,1... - From Karol A. Penson, penson(AT)lptl.jussieu.fr, March 10, 2003
Sum of consecutive combinatorial differences whose result gives (2*n)! for its numerator and 2^n for its denominator and which is the last coefficient for the lines presented in the table of sequence A087127. That is, a(n) = Sum_{i=1..n} [ C(2*n-2,2*i-2)*C(2*n-2*i+2,2*n-2*i)^(n-1) -C(2*n-2,2*i-1)*C(2*n-2*i+1,2*n-2*i-1)^(n-1) ]. E.g. a(13)= Sum_{i=1..13} [C(24,2*i-2)*C(28-2*i,26-2*i)^12 -C(24,2*i-1)*C(27-2*i,25-2*i)^12 ] = 24!/2^12 = 4!!/2^12 = 151476660579404160000 - Andre F. Labossiere (boronali(AT)laposte.net), Mar 29 2004
Number of permutations of [2n] with no increasing runs of odd length. Example: a(2)=6 because we have 1234, 13/24, 14/23, 23/14, 24/13 and 34/12 (runs separated by slashes). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 29 2004
This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant and the pairs are distinguishable. When the pairs are not distinguishable, see A001147 and A132101. For example, there are 6 ways of arranging 2 pairs [1,1], [2,2]: { [1122], [1212], [1221], [2211], [2121], [2112] } - Ross Drewe (rd(AT)labyrinth.net.au), Mar 16 2008
n married couples are seated in a row so that every wife is to the left of her husband. The recurrence a[n+1]= a[n]*((2n+1) + Binomial[2n+1,2]) conditions on whether the (n+1)st couple is seated together or seperated by at least one other person. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jun 10 2009]
a(n) is the number of functions f:[2n]->[n] such that the preimage of {y} has cardinality 2 for every y in [n]. Note that [k] denotes the set {1,2,...,k} and [0] denotes the empty set. [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]
a(n) is also the number of n by 2n (0,1)-matrices with row sum 2 and column sum 1. [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Feb 12 2010]
Number of ways that 2n people of different heights can be arranged (for a photograph) in two rows of equal length so that every person in the front row is shorter than the person immediately behind them in the back row.
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REFERENCES
| G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
Shanzhen Gao, Sequences Arising from Integer Matrix Enumeration (in preparation) [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Feb 12 2010]
Gao, Shanzhen, and Matheis, Kenneth, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
S. A. Joffe, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
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).
C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..100
D. Walsh, Counting integer functions with size-2 preimage constraints, (preprint). [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]
Eric Weisstein's World of Mathematics, Lattice Path
Index to divisibility sequences
Index entries for related partition-counting sequences
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FORMULA
| E.g.f.: 1/(1-x^2/2) (with interpolating zeros). - Paul Barry (pbarry(AT)wit.ie), May 26 2003
A000680(n) = Polygorial(n, 6) = A000142(n)/A000079(n)*A001813(n) = n!/2^n*product(4*i+2, i=0..n-1) = n!/2^n*4^n*pochhammer(1/2, n) = GAMMA(2*n+1)/2^n - Daniel Dockery (peritus(AT)gmail.com) Jun 13, 2003
For even n, a(n)=binomial(2n,n)*[a(n/2)]^2. For odd n, a(n)=binomial(2n,n+1)*a(n/2+.5)*a(n/2-.5). For positive n, a(n)=binomial(2n,2)*a(n-1) with a(0)=1. [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]
a(n) = Product_i=1...n Binomial(2i,2)
a(n) = a(n-1)* Binomial(2n,2)
Contribution from Peter Bala, 21 Feb 2011 (Start)
a(n) = product {k = 0..n-1} (T(n)-T(k)),where T(n) = n*(n+1)/2 is the n-th triangular number.
Compare with n! = product {k = 0..n-1} (n-k).
Thus we may view a(n) as a generalized factorial function associated with the triangular numbers A000217. Cf. A010050. The corresponding generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645. Also cf. A186432.
a(n) = n*(n+n-1)*(n+n-1+n-2)*...*(n+n-1+n-2+...+1).
For example, a(5) = 5*(5+4)*(5+4+3)*(5+4+3+2)*(5+4+3+2+1) = 113400. (End).
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EXAMPLE
| For n=2, a(2)=6 since there are 6 functions f:[4]->[2] with size 2 preimages for both {1} and {2}. In this case, there are binomial(4,2)=6 ways to choose the 2 elements of [4] f maps to {1} and the 2 elements of [4] that f maps to {2}. [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]
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MAPLE
| A000680 := n->(2*n)!/(2^n);
a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*(2*n-1)*n od: seq(a[n], n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008
seq(product(binomial(2*n-2*k, 2), k=0..n-1), n=0..16); [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]
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MATHEMATICA
| Table[Product[Binomial[2 i, 2], {i, 1, n}], {n, 0, 16}]
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PROG
| (PARI) a(n)=if(n<0, 0, (2*n)!/2^n)
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CROSSREFS
| Cf. A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.
Sequence in context: A177288 A177289 A177290 * A013297 A077370 A095864
Adjacent sequences: A000677 A000678 A000679 * A000681 A000682 A000683
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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