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A079484
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a(n) = (2n-1)!!*(2n+1)!!, where the double factorial is A000165.
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8
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1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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
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REFERENCES
| M. Bóna, A walk through combinatorics, World Scientific, 2006.
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LINKS
| Eric Weisstein's World of Mathematics, Struve function
G.-N. Han and C. Krattenthaler, Rectangular Scott-type permanents
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FORMULA
| a(n+1) = (4n^2-1)*a(n).
E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry (pbarry(AT)wit.ie), May 26 2003
(2n-1)! * C(2n-2, n-1) / 2^(2n-2). - R. 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. [From Olivier GERARD (olivier.gerard(AT)gmail.com), 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)! [From Vladimir Kruchinin kru(AT).ie.tusur.ru Apr 22 2011]
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EXAMPLE
| 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. [From Olivier GERARD (olivier.gerard(AT)gmail.com), May 20 2009]
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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 */
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CROSSREFS
| Cf. A001818, A000165.
Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments. [From Olivier GERARD (olivier.gerard(AT)gmail.com), May 20 2009]
Equals the row sums of A162005. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009]
Sequence in context: A144949 A144950 A144951 * A012494 A012780 A072503
Adjacent sequences: A079481 A079482 A079483 * A079485 A079486 A079487
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KEYWORD
| nonn,changed
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 17 2003
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EXTENSIONS
| Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004
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