login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882. 16
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

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-November-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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 09:47 EDT 2018. Contains 316319 sequences. (Running on oeis4.)