%I #170 Jun 23 2024 21:43:16
%S 1,3,45,1575,99225,9823275,1404728325,273922023375,69850115960625,
%T 22561587455281875,9002073394657468125,4348001449619557104375,
%U 2500100833531245335015625,1687568062633590601135546875,1321365793042101440689133203125
%N a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.
%C a(n) is the 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. - Adapted to offset 0, _Rainer Rosenthal_, Jun 19 2024
%C In the following two comments on the calculation of the terms using permanents, offset 1 is assumed. In the corresponding PARI code, this is implemented with offset 0. - _Hugo Pfoertner_, Jun 23 2024
%C (-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.
%C 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
%C a(n) is the number of permutations in S_{2n+1} in which all cycles have odd length. - _José H. Nieto S._, Jan 09 2012
%C Number of 3-bundled increasing bilabeled trees with 2n labels. - _Markus Kuba_, Nov 18 2014
%C 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
%D Miklós Bóna, A walk through combinatorics, World Scientific, 2006.
%H Alois P. Heinz, <a href="/A079484/b079484.txt">Table of n, a(n) for n = 0..224</a>
%H Cyril Banderier, Markus Kuba, and Michael Wallner, <a href="https://arxiv.org/abs/2103.03751">Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions</a>, arXiv:2103.03751 [math.PR], 2021.
%H Guo-Niu Han and Christian Krattenthaler, <a href="http://arXiv.org/abs/math.RA/0003072">Rectangular Scott-type permanents</a>, arXiv:math/0003072 [math.RA], 2000.
%H Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 17 Nov 2014.
%H MathOverflow, <a href="https://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1/401540#401540">Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?</a>, answer by Tom Copeland posted in Aug 2021.
%H Henryk Minc, <a href="http://dx.doi.org/10.1016/0024-3795(79)90128-9">On a conjecture of R. F. Scott (1881)</a>, Linear Algebra Appl., Vol. 28 (1979), pp. 141-153.
%H Theodoros Theodoulidis, <a href="http://dx.doi.org/10.3311/PPee.7894">On the Closed-Form Expression of Carson’s Integral</a>, Period. Polytech. Elec. Eng. Comp. Sci., Vol. 59, No. 1 (2015), pp. 26-29.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StruveFunction.html">Struve function</a>.
%F D-finite with recurrence a(n) = (4*n^2 - 1) * a(n-1) for all n in Z.
%F a(n) = A001147(n)*A001147(n+1).
%F E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - _Paul Barry_, May 26 2003
%F a(n) = (2n+1)! * C(2n, n) / 2^(2n). - _Ralf Stephan_, Mar 22 2004.
%F Alternatingly signed values have e.g.f. sqrt(1+x^2).
%F 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
%F a(n) = -2^(2*n-1)*exp(i*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - _Gerry Martens_, Mar 07 2011
%F 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
%F 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
%F a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - _Jean-François Alcover_, Jul 20 2015
%F Limit_{n->oo} 4^n*(n!)^2/a(n) = Pi/2. - _Daniel Suteu_, Feb 05 2017
%F From _Michael Somos_, May 04 2017: (Start)
%F a(n) = (2*n + 1) * A001818(n).
%F E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)).
%F Given e.g.f. A(x) = y, then x * y' = y + y^3.
%F a(n) = -1 / a(-1-n) for all n in Z.
%F 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)
%F a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - _Alois P. Heinz_, Jul 12 2018
%F From _Amiram Eldar_, Mar 18 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1 + L_1(1)*Pi/2, where L is the modified Struve function.
%F Sum_{n>=0} (-1)^n/a(n) = 1 - H_1(1)*Pi/2, where H is the Struve function. (End)
%e G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ...
%e M(5) =
%e [1, 2, 3, 1, 5]
%e [1, 2, 2, 4, 5]
%e [1, 3, 3, 4, 5]
%e [4, 2, 3, 4, 5]
%e [1, 2, 3, 4, 5].
%e Integral_{x=0..oo} x^3*BesselK(1, sqrt(x)) = 1575*Pi. - _Olivier Gérard_, May 20 2009
%p a:= n-> (d-> d(2*n-1)*d(2*n+1))(doublefactorial):
%p seq(a(n), n=0..15); # _Alois P. Heinz_, Jan 30 2013
%p # second Maple program:
%p A079484 := n-> LinearAlgebra[Determinant](Matrix(2*n+1, (i, j)-> `if`(i+j=2*n+1, j, i))): seq(A079484(n), n=0..14); # _Rainer Rosenthal_, Jun 18 2024
%t a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Jan 30 2013 *)
%o (PARI) /* Formula using the zeta function and a log integral:*/
%o L(n)= intnum(t=0, 1, log(1-1/t)^n);
%o Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n);
%o a(m)={my(n=m+1);round(real(-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n))))};
%o /* _Gerry Martens_, Mar 07 2011, adapted to offset 0 by _Hugo Pfoertner_, Jun 19 2024 */
%o (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
%o (PARI) {a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* _Michael Somos_, May 04 2017 */
%o (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 */
%o (PARI) \\ using the Pochhammer symbol
%o a(n) = {my(P(x,k)=gamma(x+k)/gamma(x)); 4^n*round(P(1/2,n)*P(3/2,n))} \\ _Hugo Pfoertner_, Jun 20 2024
%o (PARI) \\ Scott's (1881) method
%o a(n) = {my(m=2*n+1, X = polroots(x^m-1), Y = polroots(x^m+1), M = matrix(m, m, i, j, 1/(X[i]-Y[j]))); (-1)^n * round(2^m * real(matpermanent(M)))}; \\ _Hugo Pfoertner_, Jun 23 2024
%Y Cf. A001818, A000165.
%Y Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
%Y Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments.
%Y Equals the row sums of A162005.
%Y Cf. A316728.
%Y Diagonal elements of A306364 in even-numbered rows.
%K nonn
%O 0,2
%A _Benoit Cloitre_, Jan 17 2003
%E Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004