0,3

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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

S. A. Joffe, Quart. J. Pure Appl. Math. 47 (1914), 103-126.

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.

T. D. Noe, Table of n, a(n) for n=0..100

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for related partition-counting sequences

Counting integer functions with size-2 preimage constraints, Dennis Walsh (preprint). [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009]

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]

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]

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]

(PARI) a(n)=if(n<0, 0, (2*n)!/2^n)

Cf. A084939, A084940, A084941, A084942, A084943, A084944.

Cf. A087127.

Cf. A001147, A132101.

Sequence in context: A001499 A147630 A132467 this_sequence A013297 A095864 A006151

Adjacent sequences: A000677 A000678 A000679 this_sequence A000681 A000682 A000683

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).