A134372 -id:A134372 - cp's OEIS frontend

A000165 Double factorial of even numbers: (2n)!! = 2^n*n!.

1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000, 1678343852714360832000, 63777066403145711616000

Offset: 0

N. J. A. Sloane

a(n) is also the size of the automorphism group of the graph (edge graph) of the n-dimensional hypercube and also of the geometric automorphism group of the hypercube (the two groups are isomorphic). This group is an extension of an elementary Abelian group (C_2)^n by S_n. (C_2 is the cyclic group with two elements and S_n is the symmetric group.) - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001
Then a(n) appears in the power series: sqrt(1+sin(y)) = Sum_{n>=0} (-1)^floor(n/2)*y^(n)/a(n) and sqrt((1+cos(y))/2) = Sum_{n>=0} (-1)^n*y^(2n)/a(2n). - Benoit Cloitre, Feb 02 2002
Appears to be the BinomialMean transform of A001907. See A075271. - John W. Layman, Sep 28 2002
Number of n X n monomial matrices with entries 0, +-1.
Also number of linear signed orders.
Define a "downgrade" to be the permutation d which places the items of a permutation p in descending order. This note concerns those permutations that are equal to their double-downgrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of double-downgrading permutations of order 2n and 2n+1. - Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003
a(n) = (Integral_{x=0..Pi/2} cos(x)^(2*n+1) dx) where the denominators are b(n) = (2*n)!/(n!*2^n). - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004
1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 + ... = sqrt(1+sin(x)).
a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - Reinhard Zumkeller, Jan 14 2006
a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is j for iDavid Callan, Sep 25 2006
a(n) is the number of increasing plane trees with n+1 edges. (In a plane tree, each subtree of the root is an ordered tree but the subtrees of the root may be cyclically rotated.) Increasing means the vertices are labeled 0,1,2,...,n+1 and each child has a greater label than its parent. Cf. A001147 for increasing ordered trees, A000142 for increasing unordered trees and A000111 for increasing 0-1-2 trees. - David Callan, Dec 22 2006
Hamed Hatami and Pooya Hatami prove that this is an upper bound on the cardinality of any minimal dominating set in C_{2n+1}^n, the Cartesian product of n copies of the cycle of size 2n+1, where 2n+1 is a prime. - Jonathan Vos Post, Jan 03 2007
This sequence and (1,-2,0,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 29 2007
a(n) = number of permutations of the multiset {1,1,2,2,...,n,n,n+1,n+1} such that between the two occurrences of i, there is exactly one entry >i, for i=1,2,...,n. Example: a(2) = 8 counts 121323, 131232, 213123, 231213, 232131, 312132, 321312, 323121. Proof: There is always exactly one entry between the two 1s (when n>=1). Given a permutation p in A(n) (counted by a(n)), record the position i of the first 1, then delete both 1s and subtract 1 from every entry to get a permutation q in A(n-1). The mapping p -> (i,q) is a bijection from A(n) to the Cartesian product [1,2n] X A(n-1). - David Callan, Nov 29 2007
Row sums of A028338. - Paul Barry, Feb 07 2009
a(n) is the number of ways to seat n married couples in a row so that everyone is next to their spouse. Compare A007060. - Geoffrey Critzer, Mar 29 2009
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 2, 8, 48, ...) dot (1, -3, 5, -7, 9, ...).
Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, -3, 5, -7, 9) = (1, -3, 10, -56, 432). (End)
exp(x/2) = Sum_{n>=0} x^n/a(n). - Jaume Oliver Lafont, Sep 07 2009
Assuming n starts at 0, a(n) appears to be the number of Gray codes on n bits. It certainly is the number of Gray codes on n bits isomorphic to the canonical one. Proof: There are 2^n different starting positions for each code. Also, each code has a particular pattern of bit positions that are flipped (for instance, 1 2 1 3 1 2 1 for n=3), and these bit position patterns can be permuted in n! ways. - D. J. Schreffler (ds1404(AT)txstate.edu), Jul 18 2010
E.g.f. of 0,1,2,8,... is x/(1-2x/(2-2x/(3-8x/(4-8x/(5-18x/(6-18x/(7-... (continued fraction). - Paul Barry, Jan 17 2011
Number of increasing 2-colored trees with choice of two colors for each edge. In general, if we replace 2 with k we get the number of increasing k-colored trees. For example, for k=3 we get the triple factorial numbers. - Wenjin Woan, May 31 2011
a(n) = row sums of triangle A193229. - Gary W. Adamson, Jul 18 2011
Also the number of permutations of 2n (or of 2n+1) that are equal to their reverse-complements. (See the Egge reference.) Note that the double-downgrade described in the preceding comment (McDonnell) is equivalent to the reverse-complement. - Justin M. Troyka, Aug 11 2011
The e.g.f. can be used to form a generator, [1/(1-2x)] d/dx, for A000108, so a(n) can be applied to A145271 to generate the Catalan numbers. - Tom Copeland, Oct 01 2011
The e.g.f. of 1/a(n) is BesselI(0,sqrt(2*x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 09 2012
a(n) = order of the largest imprimitive group of degree 2n with n systems of imprimitivity (see [Miller], p. 203). - L. Edson Jeffery, Feb 05 2012
Row sums of triangle A208057. - Gary W. Adamson, Feb 22 2012
a(n) is the number of ways to designate a subset of elements in each n-permutation. a(n) = A000142(n) + A001563(n) + A001804(n) + A001805(n) + A001806(n) + A001807(n) + A035038(n) * n!. - Geoffrey Critzer, Nov 08 2012
For n>1, a(n) is the order of the Coxeter groups (also called Weyl groups) of types B_n and C_n. - Tom Edgar, Nov 05 2013
For m>0, k*a(m-1) is the m-th cumulant of the chi-squared probability distribution for k degrees of freedom. - Stanislav Sykora, Jun 27 2014
a(n) with 0 prepended is the binomial transform of A120765. - Vladimir Reshetnikov, Oct 28 2015
Exponential self-convolution of A001147. - Vladimir Reshetnikov, Oct 08 2016
Also the order of the automorphism group of the n-ladder rung graph. - Eric W. Weisstein, Jul 22 2017
a(n) is the order of the group O_n(Z) = {A in M_n(Z): A*A^T = I_n}, the group of n X n orthogonal matrices over the integers. - Jianing Song, Mar 29 2021
a(n) is the number of ways to tile a (3n,3n)-benzel or a (3n+1,3n+2)-benzel using left stones and two kinds of bones; see Defant et al., below. - James Propp, Jul 22 2023
a(n) is the number of labeled histories for a labeled topology with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

			The following permutations and their reversals are all of the permutations of order 5 having the double-downgrade property:
  0 1 2 3 4
  0 3 2 1 4
  1 0 2 4 3
  1 4 2 0 3
G.f. = 1 + 2*x + 8*x^2 + 48*x^3 + 384*x^4 + 3840*x^5 + 46080*x^6 + 645120*x^7 + ...

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS - Combinatorial Object Server, Generate colored permutations
R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015.
Colin Defant, Rupert Li, James Propp, and Benjamin Young, Tilings of Benzels via the Abacus Bijection, arXiv preprint, arXiv:2209.05717 [math.CO], 2022.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
G. Gordon, The answer is 2^n*n! What is the question?, Amer. Math. Monthly, 106 (1999), 636-645.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Hamed Hatami and Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles, arXiv:math/0701018 [math.CO], 2006-2009.
Jason D. Hildebrand, Differentiating Arctan(x)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136
E. Lappo and N. A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
Eugene McDonnell, Magic Squares and Permutations, APL Quote Quad 7.3 (Fall 1976).
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
G. A. Miller, Groups formed by special matrices, Bull. Amer. Math. Soc. 24 (1918), 203-206.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Graph Automorphism
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Index to divisibility sequences
Index entries for sequences related to factorial numbers

Cf. A000142 (n!), A001147 ((2n-1)!!), A032184 (2^n*(n-1)!).
Cf. A000079, A000108, A000111, A001044, A001563, A001804, A001805, A001806.
Cf. A001807, A001813, A001907, A002454, A006882, A007060, A008277, A008544.
Cf. A010050, A019774, A028326, A028338, A035038, A039683, A047053, A047055.
Cf. A047058, A047657, A075271, A084947, A084948, A084949, A092605, A133314.
Cf. A145271, A193229, A208057.
This sequence gives the row sums in A060187, and (-1)^n*a(n) the alternating row sums in A039757.
Also row sums in A028338.
Column k=2 of A329070.

a000165 n = product [2, 4 .. 2 * n]  -- Reinhard Zumkeller, Mar 28 2015

[2^n*Factorial(n): n in [0..35]]; // Vincenzo Librandi, Apr 22 2011

I:=[2,8]; [1] cat [n le 2 select I[n]  else (3*n-1)*Self(n-1)-2*(n-1)^2*Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Feb 19 2015

A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2); fi; end;
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 0)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..17); # Zerinvary Lajos, Mar 26 2008
G(x):=(1-2*x)^(-1): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..17); # Zerinvary Lajos, Apr 03 2009
A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n),n=0..10) ; # R. J. Mathar, Oct 20 2009

Table[(2 n)!!, {n, 30}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
(2 Range[0, 30])!! (* Harvey P. Dale, Jan 23 2015 *)
RecurrenceTable[{a[n] == 2 n*a[n-1], a[0] == 1}, a, {n,0,30}] (* Ray Chandler, Jul 30 2015 *)

a(n)=n!<Charles R Greathouse IV, Feb 11 2011

{a(n) = prod( k=1, n, 2*k)}; /* Michael Somos, Jan 04 2013 */

from math import factorial
def A000165(n): return factorial(n)<Chai Wah Wu, Jan 24 2023

[2^n*factorial(n) for n in range(31)] # G. C. Greubel, Jul 21 2024

E.g.f.: 1/(1-2*x).
a(n) = A001044(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (2*i+2) = 2^n*Pochhammer(1,n). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
D-finite with recurrence a(n) = 2*n * a(n-1), n>0, a(0)=1. - Paul Barry, Aug 26 2004
This is the binomial mean transform of A001907. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
a(n) = Integral_{x>=0} x^n*exp(-x/2)/2 dx. - Paul Barry, Jan 28 2008
G.f.: 1/(1-2x/(1-2x/(1-4x/(1-4x/(1-6x/(1-6x/(1-.... (continued fraction). - Paul Barry, Feb 07 2009
a(n) = A006882(2*n). - R. J. Mathar, Oct 20 2009
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) = upper left term in M^n, M = a production matrix (twice Pascal's triangle deleting the first "2", with the rest zeros; cf. A028326):
  2,  2,  0,  0,  0,  0, ...
  2,  4,  2,  0,  0,  0, ...
  2,  6,  6,  2,  0,  0, ...
  2,  8, 12,  8,  2,  0, ...
  2, 10, 20, 20, 10,  2, ...
  ... (End)
From Sergei N. Gladkovskii, Apr 11 2013, May 01 2013, May 24 2013, Sep 30 2013, Oct 27 2013: (Start)
Continued fractions:
G.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 + (2*k+2)/(1-x/(x+1/Q(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 + 2*k*x - 2*x*(k+1)/Q(k+1).
G.f.: G(0)/2 where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 - x*(4*k+2) - 4*x^2*(k+1)^2/Q(k+1).
G.f.: R(0) where R(k) = 1 - x*(2*k+2)/(x*(2*k+2)-1/(1-x*(2*k+2)/(x*(2*k+2) -1/R(k+1)))). (End)
a(n) = (2n-2)*a(n-2) + (2n-1)*a(n-1), n>1. - Ivan N. Ianakiev, Aug 06 2013
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n - 1)*a(n-1) - 2*(n - 1)^2*a(n-2) with a(1) = 2 and a(2) = 8.
The sequence b(n) = A068102(n) also satisfies this second-order recurrence. This leads to the generalized continued fraction expansion lim_{n -> oo} b(n)/a(n) = log(2) = 1/(2 - 2/(5 - 8/(8 - 18/(11 - ... - 2*(n - 1)^2/((3*n - 1) - ... ))))). (End)
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = sqrt(e) (A019774).
Sum_{n>=0} (-1)^n/a(n) = 1/sqrt(e) (A092605). (End)
Limit_{n->oo} a(n)^4 / (n * A134372(n)) = Pi. - Daniel Suteu, Apr 09 2022
a(n) = 1/([x^n] hypergeom([1], [1], x/2)). - Peter Luschny, Sep 13 2024
a(n) = Sum_{k=0..n} k!*(n-k)!*binomial(n,k)^2. - Ridouane Oudra, Jul 13 2025

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

cp's OEIS Frontend

A000165 Double factorial of even numbers: (2n)!! = 2^n*n!.

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A134375 a(n) = (n!)^4.

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A134374 a(n) = ((2n+1)!)^2.

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A161736 Denominators of the column sums of the BG2 matrix.

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A127488 a(n) = (n^2)!/(2*(n!)).

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A264153 a(n) = ((2*n)!)^2 / 2^n.

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A372184 a(n) = 2^(1-3*n)*((2*n)!)^2/n.

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A372184 a(n) = 2^(1-3n)((2*n)!)^2/n.