

A000680


a(n) = (2n)!/2^n.
(Formerly M4287 N1793)


66



1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 2375880867360000, 548828480360160000, 151476660579404160000, 49229914688306352000000, 18608907752179801056000000, 8094874872198213459360000000, 4015057936610313875842560000000
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OFFSET

0,3


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)mydeja.com), Apr 20 2001
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
a(n) is also the number of lattice paths in the ndimensional lattice [0..2]^n.  T. D. Noe, Jun 06 2002
Representation as the nth moment of a positive function on the positive halfaxis: a(n) = Integral_{x>=0} (x^n*exp(sqrt(2*x))/sqrt(2*x)), n=0,1,...  Karol A. Penson, Mar 10 2003
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
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
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
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
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
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.
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
a(n) is also the 2*n th difference of npowers 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
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
Number of ways to put socks and shoes on an nlegged animal, if a sock must be put on before a shoe.  Daniel Bishop, Jan 29 2018


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 AddisonWesley, Reading, MA, 1962, Vol. 1, p. 112.
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 FortyFirst Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 4553.
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. 95106; Amer. Math. Soc., Providence, R.I.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
R. Florez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12(1) (2012), 8396.
M. Ghebleh, Antichains of (0, 1)matrices through inversions, Linear Algebra and its Applications 458 (2014), 503511.
S. A. Joffe, Calculation of the first thirtytwo Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103126. [Accessible only in the USA through the Hathi Trust Digital Library.]
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
J.C. Novelli and J.Y. Thibon, Hopf Algebras of mpermutations,(m+1)ary trees, and mparking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 20062007.
Dennis Walsh, Counting integer functions with size2 preimage constraints, preprint.
Eric Weisstein's World of Mathematics, Lattice Path.
Index to divisibility sequences
Index entries for related partitioncounting sequences


FORMULA

E.g.f.: 1/(1  x^2/2) (with interpolating zeros).  Paul Barry, May 26 2003
a(n) = polygorial(n, 6) = (A000142(n)/A000079(n))*A001813(n) = (n!/2^n)*Product_{i=0..n1} (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
a(n) = A087127(n,2*n) = Sum_{i=0..2*n} (1)^(2*ni)*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*kj+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]
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((n1)/2). For positive n, a(n) = binomial(2n, 2)*a(n1) with a(0) = 1.  Dennis P. Walsh, Nov 17 2009
a(n) = Product_{i=1..n} binomial(2i, 2).
a(n) = a(n1)*binomial(2n, 2).
From Peter Bala, Feb 21 2011: (Start)
a(n) = Product_{k = 0..n1} (T(n)  T(k)), where T(n) = n*(n + 1)/2 is the nth triangular number.
Compare with n! = Product_{k = 0..n1} (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(nk)) are triangle A086645. Also cf. A186432.
a(n) = n*(n + n1)*(n + n1 + n2)*...*(n + n1 + n2 + ... + 1).
For example, a(5) = 5*(5+4)*(5+4+3)*(5+4+3+2)*(5+4+3+2+1) = 113400. (End).
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, 1step).  Sergei N. Gladkovskii, Oct 28 2012
a(n) = n!*(product of the first n odd integers).  Dennis P. Walsh, Nov 28 2012
a(0) = 1, a(n) = a(n1)*T(2*n1), where T(n) is the nth triangular number. For example: a(4) = a(3)*T(7) = 90*28 = 2520.  Enric Reverter i Bigas, Jun 24 2013
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
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0) 1/a(n) = cosh(sqrt(2)).
Sum_{n>=0) (1)^n/a(n) = cos(sqrt(2)). (End)
Dfinite with recurrence a(n) n*(2*n1)*a(n1)=0.  R. J. Mathar, Jan 28 2022
a(n) = n *A007019(n1), n>0.  R. J. Mathar, Jan 28 2022


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}.  Dennis P. Walsh, Nov 17 2009


MAPLE

A000680 := n>(2*n)!/(2^n);
a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=a[n1]*(2*n1)*n od: seq(a[n], n=0..16); # Zerinvary Lajos, Mar 08 2008
seq(product(binomial(2*n2*k, 2), k=0..n1), n=0..16); # Dennis P. Walsh, Nov 17 2009


MATHEMATICA

Table[Product[Binomial[2 i, 2], {i, 1, n}], {n, 0, 16}]
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 *)
Table[(2n)!/2^n, {n, 0, 20}] (* Harvey P. Dale, Sep 21 2020 *)


PROG

(PARI) a(n) = (2*n)! / 2^n


CROSSREFS

Cf. A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.
A diagonal of the triangle in A241171.
Main diagonal of A267479, row sums of A267480.
Row n=2 of A089759.
Column n=2 of A187783.
Even bisection of column k=0 of A097591.
Sequence in context: A177288 A177289 A177290 * A013297 A232979 A218758
Adjacent sequences: A000677 A000678 A000679 * A000681 A000682 A000683


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



