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

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

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

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

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

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

Cf. A087127.

Cf. A001147, A132101.

Adjacent sequences: A000677 A000678 A000679 this_sequence A000681 A000682 A000683

Sequence in context: A004996 A001499 A132467 this_sequence A013297 A095864 A006151

nonn,easy

njas