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 A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2. (Formerly M4669 N1997) 90
 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 From Zhi-Wei Sun, Jun 26 2022: (Start) Conjecture 1：For any primitive 2n-th root zeta of unity, the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} coincides with a(n) = ((2n-1)!!)^2, where m(j,k) is (1+zeta^(j-k))/(1-zeta^(j-k)) if j is not equal to k, and 1 otherwise. The determinant of [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n-1)!!)^2/(2n-1) by Han Wang and Zhi-Wei Sun in 2022. Conjecture 2：Let p be an odd prime. Then the permanent of (p-1) X (p-1) matrix [f(j,k)]_{j,k=1..p-1} is congruent to a((p-1)/2) = ((p-2)!!)^2 modulo p^2, where f(j,k) is (j+k)/(j-k) if j is not equal to k, and f(j,k) = 1 otherwise. (End) REFERENCES John 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). Richard 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. Harry Crane and Peter McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, Vol. 52, No. 3 (2015), pp. 622-635. John Engbers, David Galvin, and 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. Terence Tao, A differentiation identity. Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022. Eric Weisstein's World of Mathematics, Struve function. Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021. FORMULA a(n) = A001147(n)^2. a(n) = A111595(2*n, 0). a(n) = (2*n-1)!*Sum_{k=0..n-1} binomial(2*k,k)/4^k, n >= 1. - Wolfdieter Lang, Aug 23 2005 arcsinh(x) = Sum_{n>=1} (-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!. - James R. Buddenhagen, Mar 24 2009 From Karol A. Penson, Oct 21 2009: (Start) G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 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) = Integral_{x>=0} x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)). This solution is unique. (End) D-finite with recurrence: 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 From Robert FERREOL, Jul 30 2020: (Start) a(n) = ((2*n)!/4^n)*binomial(2*n,n). a(n) = (2*n-1)!*Sum_{k=0..n-1} a(k)/(2*k)!, n >= 1. a(n) = A184877(2*n-1) for n>=1. (End) From Amiram Eldar, Mar 18 2022: (Start) Sum_{n>=0} 1/a(n) = 1 + L_0(1)*Pi/2, where L is the modified Struve function (see A197037). Sum_{n>=0} (-1)^n/a(n) = 1 - H_0(1)*Pi/2, where H is the Struve function. (End) 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 Cf. A001147, A002454, A111595, A197037. Bisection of A012248. Right-hand column 1 in triangle A008956. Sequence in context: A251579 A128492 A294971 * A095363 A138564 A285985 Adjacent sequences:  A001815 A001816 A001817 * A001819 A001820 A001821 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009 STATUS approved

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Last modified September 26 01:31 EDT 2022. Contains 356986 sequences. (Running on oeis4.)