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 A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882. 19
 1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875, 1321365793042101440689133203125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = determinant of M(2n-1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k m(i,j)=i otherwise. (-1)^n*a(n)/2^(2n-1) is the permanent of the (m X m) matrix {1/(x_i-y_j), 1<=i<=m, 1<=j<=m}, where x_1,x_2,...,x_m are the zeros of x^m-1 and y_1,y_2,...,y_m the zeros of y^m+1 and m=2n-1. a(n) = number of permutations in S_{2n+1} in which all cycles have odd length. - José H. Nieto S., Jan 09 2012 In 1881, R. F. Scott posed a conjecture that the absolute value of permanent of square matrix with elements a(i,j)= (x_i - y_j)^(-1), where x_1,...,x_n are roots of x^n=1, while y_1,...,y_n are roots of y^n=-1, equals a((n-1)/2)/2^n, if n>=1 is odd, and 0, if n>=2 is even. After a century (in 1979), the conjecture was proved by H. Minc. - Vladimir Shevelev, Dec 01 2013 Number of 3-bundled increasing bilabeled trees with 2n labels. - Markus Kuba, Nov 18 2014 a(n) is the number of rooted, binary, leaf-labeled topologies with 2n+2 leaves that have n+1 cherry nodes. - Noah A Rosenberg, Feb 12 2019 REFERENCES M. Bóna, A walk through combinatorics, World Scientific, 2006. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..224 G.-N. Han and C. Krattenthaler, Rectangular Scott-type permanents, arXiv:math/0003072 [math.RA], 2000. Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 17 Nov 2014. H. Minc, On a conjecture of R. F. Scott (1881), Linear Algebra Appl., 28(1979), 141-153. Theodoros Theodoulidis, On the Closed-Form Expression of Carson’s Integral, Period. Polytech. Elec. Eng. Comp. Sci., Vol. 59, No. 1 (2015), pp. 26-29. Eric Weisstein's World of Mathematics, Struve function FORMULA a(n) = (4*n^2 - 1) * a(n-1) for all n in Z. a(n) = A001147(n)*A001147(n+1). E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry, May 26 2003 a(n) = (2n+1)! * C(2n, n) / 2^(2n). - Ralf Stephan, Mar 22 2004. Alternatingly signed values have e.g.f. sqrt(1+x^2). a(n) is the value of the n-th moment of : 1/Pi BesselK(1, sqrt(x)) on the positive part of the real line. - Olivier Gérard, May 20 2009 a(n) = -2^(2*n-1)*exp(I*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - Gerry Martens, Mar 07 2011 E.g.f. (odd powers) tan(arcsin(x))=sum(n>=0, (2n-1)!!*(2n+1)!!*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Apr 22 2011 G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((2*k+2)^2-1)/(1-x/(x - 1/G(k+1) )); ( continued fraction ). - Sergei N. Gladkovskii, Jan 15 2013 a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - Jean-François Alcover, Jul 20 2015 Lim_{n->infinity} 4^n*(n!)^2/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017 From Michael Somos, May 04 2017: (Start) a(n) = (2*n + 1) * A001818(n). E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)). Given e.g.f. A(x) = y, then x * y' = y + y^3. a(n) = -1 / a(-1-n) for all n in Z. 0 = +a(n)*(+288*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)) for all n in Z. (End) a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - Alois P. Heinz, Jul 12 2018 EXAMPLE G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ... M(5) = [1, 2, 3, 1, 5] [1, 2, 2, 4, 5] [1, 3, 3, 4, 5] [4, 2, 3, 4, 5] [1, 2, 3, 4, 5]. int( x^3 besselK(1, sqrt(x)), x=0..infty) = 1575 Pi. - Olivier Gérard, May 20 2009 MAPLE A079484:= n-> doublefactorial(2*n-1)*doublefactorial(2*n+1): seq(A079484(n), n=0..15);  # Alois P. Heinz, Jan 30 2013 MATHEMATICA a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 30 2013 *) PROG (PARI) /* Formula using the zeta function and a log integral:*/ L(n)=  intnum(t=0, 1, log(1-1/t)^n); Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n); a(n)=-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n)); /* Gerry Martens, Mar 07 2011 */ (MAGMA)  I:=[1, 3]; [n le 2 select I[n] else (4*n^2-8*n+3)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Nov 18 2014 (PARI) {a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* Michael Somos, May 04 2017 */ (PARI) {a(n) = if( n<0, -1 / self()(-1-n), my(m = 2*n + 1); m! * polcoeff( x / sqrt( 1 - x^2 + x * O(x^m) ), m))}; /* Michael Somos, May 04 2017 */ CROSSREFS Cf. A001818, A000165. Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909. Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments. Equals the row sums of A162005. Cf. A316728. Diagonal elements of A306364 in even-numbered rows. Sequence in context: A144949 A144950 A144951 * A012494 A012780 A072503 Adjacent sequences:  A079481 A079482 A079483 * A079485 A079486 A079487 KEYWORD nonn AUTHOR Benoit Cloitre, Jan 17 2003 EXTENSIONS Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004 STATUS approved

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Last modified October 15 00:04 EDT 2019. Contains 328025 sequences. (Running on oeis4.)