Search: id:A000680 Results 1-1 of 1 results found. %I A000680 M4287 N1793 %S A000680 1,1,6,90,2520,113400,7484400,681080400,81729648000,12504636144000, %T A000680 2375880867360000,548828480360160000,151476660579404160000,49229914688306352000000, %U A000680 18608907752179801056000000,8094874872198213459360000000,4015057936610313875842560000000 %N A000680 (2n)!/2^n. %C A000680 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 %C A000680 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 %C A000680 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 %C A000680 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 %C A000680 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 %C A000680 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 %C A000680 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 %C A000680 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] %C A000680 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] %D A000680 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000680 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000680 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998. %D A000680 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. %D A000680 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. %D A000680 S. A. Joffe, Quart. J. Pure Appl. Math. 47 (1914), 103-126. %D A000680 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. %H A000680 T. D. Noe, Table of n, a(n) for n=0..100 %H A000680 A. F. Labossiere, Sobalian Coefficients. %H A000680 A. F. Labossiere, Miscellaneous. %H A000680 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000680 Index entries for related partition-counting sequences %H A000680 Counting integer functions with size-2 preimage constraints, Dennis Walsh (preprint). [From Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 17 2009] %F A000680 E.g.f.: 1/(1-x^2/2) (with interpolating zeros). - Paul Barry (pbarry(AT)wit.ie), May 26 2003 %F A000680 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 %F A000680 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] %e A000680 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] %p A000680 A000680 := n->(2*n)!/(2^n); %p A000680 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 %p A000680 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] %o A000680 (PARI) a(n)=if(n<0,0,(2*n)!/2^n) %Y A000680 Cf. A084939, A084940, A084941, A084942, A084943, A084944. %Y A000680 Cf. A087127. %Y A000680 Cf. A001147, A132101. %Y A000680 Sequence in context: A001499 A147630 A132467 this_sequence A013297 A095864 A006151 %Y A000680 Adjacent sequences: A000677 A000678 A000679 this_sequence A000681 A000682 A000683 %K A000680 nonn,easy,new %O A000680 0,3 %A A000680 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.002 seconds