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A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.
(Formerly M4669 N1997)
36
1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625, 45564337691106946230659765625, 38319607998220941779984862890625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).

Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic, Aug 10 2007

a(n) is the sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007

arcsinh(x) = sum((-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!, n=1..infinity). - James R. Buddenhagen, Mar 24 2009

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

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).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..50

David Callan and Emeric Deutsch, The Run Transform, arXiv preprint arXiv:1112.3639 [math.CO], 2011.

H. Crane and P. McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, 52(3), 2015.

John Engbers, David Galvin, Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 6.

IBM, "Ponder This" puzzle for June 2009. [From Vladeta Jovovic, Jul 26 2009]

John Riordan and N. J. A. Sloane, Correspondence, 1974

T. Tao, A differentiation identity

Eric Weisstein's World of Mathematics, Struve function

Index to divisibility sequences

FORMULA

a(n) = (2*n-1)!*sum(binomial(2*k,k)/4^k,k=0..n-1), n>=1. - Wolfdieter Lang, Aug 23 2005

From Karol A. Penson, Oct 21 2009: (Start)

G.f.:sum(a(n)*x^n/(n!)^2,n=0..infinity)=2*EllipticK(2*sqrt(x))/Pi.

Asymptotically: a(n)=(2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/(((1/n)^n)^2*(exp(n))^2), n->infinity.

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:

a(n)=int(x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)),x=0..infinity), n=0,1... .

This solution is unique.

(End)

a(0) = 1, a(n) = (2*n-1)^2*a(n-1), n>0.

a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos, Jul 03 2002

(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002

-arccos(x)+ pi/2 = x + x^3/3! + 9 x^5/5! + 225 x^7/7! + 11205 x^9/9! + ... - Tom Copeland, Oct 23 2008

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) =  1 - (4*k^2+4*k+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013

a(n) = det(V(i+1,j), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices. - Mircea Merca, Apr 04 2013

a(n) = (1+x^2)^(n+1/2) * (d/dx)^(2*n) (1+x^2)^(n-1/2).  See Tao link. - Robert Israel, Jun 04 2015

a(n) = 4^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017

0 = a(n)*(+384*a(n+2) - 60*a(n+3) + a(n+4)) + a(n+1)*(-36*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) and a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 06 2017

EXAMPLE

Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.

G.f. = 1 + x + 9*x^2 + 225*x^3 + 11025*x^4 + 893025*x^5 + 108056025*x^6 + ...

MAPLE

a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2), k=1..2*m) end; # Peter Luschny, Jun 01 2009

MATHEMATICA

FoldList[Times, 1, Range[1, 25, 2]]^2 (* or *) Join[{1}, (Range[1, 29, 2]!!)^2] (* Harvey P. Dale, Jun 06 2011, Apr 10 2012 *)

Table[((2 n - 1)!!)^2, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)

PROG

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

(PARI) {a(n) = if( n<0, 1 / a(-n), sqr((2*n)! / (n! * 2^n)))}; /* Michael Somos, Jan 06 2017 */

(MAGMA) DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^2: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2017

CROSSREFS

a(n) = A001147(n)^2. Cf. A002454.

Bisection of A012248.

Right-hand column 1 in triangle A008956.

a(n) = A111595(2*n, 0).

Sequence in context: A079727 A251579 A128492 * A095363 A138564 A285985

Adjacent sequences:  A001815 A001816 A001817 * A001819 A001820 A001821

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009

STATUS

approved

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Last modified September 25 13:09 EDT 2017. Contains 292486 sequences.