|
%I M4287 N1793 #187 May 13 2022 16:09:51
%S 1,1,6,90,2520,113400,7484400,681080400,81729648000,12504636144000,
%T 2375880867360000,548828480360160000,151476660579404160000,
%U 49229914688306352000000,18608907752179801056000000,8094874872198213459360000000,4015057936610313875842560000000
%N a(n) = (2n)!/2^n.
%C 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 a(n) is also the constant term in the product: Product_{1<=i, j<=n, i!=j} (1 - x_i/x_j)^2. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 12 2002
%C a(n) is also the number of lattice paths in the n-dimensional lattice [0..2]^n. - _T. D. Noe_, Jun 06 2002
%C Representation as the n-th moment of a positive function on the positive half-axis: a(n) = Integral_{x>=0} (x^n*exp(-sqrt(2*x))/sqrt(2*x)), n=0,1,... - _Karol A. Penson_, Mar 10 2003
%C 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_, Aug 29 2004
%C 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_, Mar 16 2008
%C 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)*((2*n + 1) + binomial(2*n+1, 2)) conditions on whether the (n+1)st couple is seated together or separated by at least one other person. - _Geoffrey Critzer_, Jun 10 2009
%C 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. - _Dennis P. Walsh_, Nov 17 2009
%C a(n) is also the number of n X 2n (0,1)-matrices with row sum 2 and column sum 1. - _Shanzhen Gao_, Feb 12 2010
%C 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.
%C a(n) is the number of functions f:[n]->[n^2] such that, if floor((f(x))^.5) = floor((f(y))^.5), then x = y. For example, with n = 4, the range of f consists of one element from each of the four sets {1,2,3}, {4,5,6,7,8}, {9,10,11,12,13,14,15}, and {16}. Hence there are 1*3*5*7 = 105 ways to choose the range for f, and there are 4! ways to injectively map {1,2,3,4} to the four elements of the range. Thus there are 105*24 = 2520 such functions. Note also that a(n) = n!*(product of the first n odd numbers). - _Dennis P. Walsh_, Nov 28 2012
%C a(n) is also the 2*n th difference of n-powers of A000217 (triangular numbers). For example a(2) is the 4th difference of the squares of triangular numbers. - _Enric Reverter i Bigas_, Jun 24 2013
%C a(n) is the multinomial coefficient (2*n) over (2, 2, 2, ..., 2) where there are n 2's in the last parenthesis. It is therefore also the number of words of length 2n obtained with n letters, each letter appearing twice. - _Robert FERREOL_, Jan 14 2018
%C Number of ways to put socks and shoes on an n-legged animal, if a sock must be put on before a shoe. - _Daniel Bishop_, Jan 29 2018
%D G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998.
%D 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 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 Shanzhen Gao and Kenneth Matheis, 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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 T. D. Noe, <a href="/A000680/b000680.txt">Table of n, a(n) for n = 0..100</a>
%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.
%H R. Florez and L. Junes, <a href="http://www.emis.de/journals/INTEGERS/papers/l50/l50.Abstract.html">A relation between triangular numbers and prime numbers</a>, Integers 12(1) (2012), 83-96.
%H M. Ghebleh, <a href="https://doi.org/10.1016/j.laa.2014.06.021">Antichains of (0, 1)-matrices through inversions</a>, Linear Algebra and its Applications 458 (2014), 503-511.
%H S. A. Joffe, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.$b417567&view=1up&seq=117">Calculation of the first thirty-two Eulerian numbers from central differences of zero</a>, Quart. J. Pure Appl. Math. 47 (1914), 103-126. [Accessible only in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H Peter D. Loly and Ian D. Cameron, <a href="https://arxiv.org/abs/2008.11020">Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy</a>, arXiv:2008.11020 [math.HO], 2020.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H Dennis Walsh, <a href="http://www.mtsu.edu/~dwalsh/PREIMAGE.pdf">Counting integer functions with size-2 preimage constraints</a>, preprint.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticePath.html">Lattice Path</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F E.g.f.: 1/(1 - x^2/2) (with interpolating zeros). - _Paul Barry_, May 26 2003
%F a(n) = polygorial(n, 6) = (A000142(n)/A000079(n))*A001813(n) = (n!/2^n)*Product_{i=0..n-1} (4*i + 2) = (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 a(n) = A087127(n,2*n) = Sum_{i=0..2*n} (-1)^(2*n-i)*binomial(2*n, i)*binomial(i+2, 2)^n. Let T(n,k,j) = ((n - k + j)*(2*n - 2*k + 1))^n*binomial(2*n, 2*k-j+1) then a(n) = Sum{k=0..n} (T(n,k,1) - T(n,k,0)). For example a(12) = A087127(12,24) = Sum_{k=0..12} (T(12,k,1) - T(12,k,0)) = 24!/2^12. - _André F. Labossière_, Mar 29 2004 [Corrected by _Jianing Song_, Jan 08 2019]
%F For even n, a(n) = binomial(2n, n)*(a(n/2))^2. For odd n, a(n) = binomial(2n, n+1)*a((n+1)/2)*a((n-1)/2). For positive n, a(n) = binomial(2n, 2)*a(n-1) with a(0) = 1. - _Dennis P. Walsh_, Nov 17 2009
%F a(n) = Product_{i=1..n} binomial(2i, 2).
%F a(n) = a(n-1)*binomial(2n, 2).
%F From _Peter Bala_, Feb 21 2011: (Start)
%F a(n) = Product_{k = 0..n-1} (T(n) - T(k)), where T(n) = n*(n + 1)/2 is the n-th triangular number.
%F Compare with n! = Product_{k = 0..n-1} (n - k).
%F 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.
%F a(n) = n*(n + n-1)*(n + n-1 + n-2)*...*(n + n-1 + n-2 + ... + 1).
%F For example, a(5) = 5*(5+4)*(5+4+3)*(5+4+3+2)*(5+4+3+2+1) = 113400. (End).
%F G.f.: 1/U(0) where U(k)= x*(2*k - 1)*k + 1 - x*(2*k + 1)*(k + 1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Oct 28 2012
%F a(n) = n!*(product of the first n odd integers). - _Dennis P. Walsh_, Nov 28 2012
%F a(0) = 1, a(n) = a(n-1)*T(2*n-1), where T(n) is the n-th triangular number. For example: a(4) = a(3)*T(7) = 90*28 = 2520. - _Enric Reverter i Bigas_, Jun 24 2013
%F E.g.f.: 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, May 10 2017
%F From _Amiram Eldar_, Jun 25 2020: (Start)
%F Sum_{n>=0) 1/a(n) = cosh(sqrt(2)).
%F Sum_{n>=0) (-1)^n/a(n) = cos(sqrt(2)). (End)
%F D-finite with recurrence a(n) -n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 28 2022
%F a(n) = n *A007019(n-1), n>0. - _R. J. Mathar_, Jan 28 2022
%e 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}. - _Dennis P. Walsh_, Nov 17 2009
%p A000680 := n->(2*n)!/(2^n);
%p 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_, Mar 08 2008
%p seq(product(binomial(2*n-2*k,2),k=0..n-1),n=0..16); # _Dennis P. Walsh_, Nov 17 2009
%t Table[Product[Binomial[2 i, 2], {i, 1, n}], {n, 0, 16}]
%t polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[6, #] &, 17, 0] (* _Robert G. Wilson v_, Dec 26 2016 *)
%t Table[(2n)!/2^n,{n,0,20}] (* _Harvey P. Dale_, Sep 21 2020 *)
%o (PARI) a(n) = (2*n)! / 2^n
%Y Cf. A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.
%Y A diagonal of the triangle in A241171.
%Y Main diagonal of A267479, row sums of A267480.
%Y Row n=2 of A089759.
%Y Column n=2 of A187783.
%Y Even bisection of column k=0 of A097591.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
|
