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%I #38 Nov 05 2016 08:12:41
%S 1,1,2,4,13,41,196,862,5489,31033,247006,1706816,16302397,133131649,
%T 1483518128,13978823146,178022175361,1901119947857,27237392830234,
%U 325091511083548,5175104637744461,68269217327545081,1195449171318970492,17272111983868905494
%N Expansion of e.g.f.: exp(x)/(1-x^2/2).
%C a(n) is also the number of permutations in S_n whose prefix transposition distance is tight with respect to Dias and Meidanis' lower bound (proof: see Fortuna). - _Anthony Labarre_, Feb 16 2009
%C From _Stanislav Sykora_, Nov 03 2016: (Start)
%C a(n) is the number of unary operators (involutions) on S_n, i.e., endomorphisms U such that U^2 is the identity mapping, including the identity itself.
%C Physics example: a particle with a half-integer spin s has a discrete spin space with 2s+1 base states which admits a(2s+1) linear unary operators (including the identity). These are important because they satisfy the operator identity exp(izU) = coz(z)+i*sin(z)*U, valid for any complex z.
%C (End)
%H Stanislav Sykora, <a href="/A087214/b087214.txt">Table of n, a(n) for n = 0..199</a>
%H Zanoni Dias and Joao Meidanis, <a href="http://dx.doi.org/10.1007/3-540-45735-6_7">Sorting by Prefix Transpositions</a>, Proceedings of the Ninth International Symposium on String Processing and Information Retrieval (SPIRE), 2002, 65-76, vol. 2476 of Lecture Notes in Computer Science, Springer-Verlag. [_Anthony Labarre_, Feb 16 2009]
%H Zanoni Dias, Vinicius Fortuna and Joao Meidanis, <a href="http://www.ic.unicamp.br/~meidanis/PUB/Mestrado/2003-Fortuna/prefix-2004-07-30.pdf">Sorting by Prefix Transpositions</a>, 2004.
%H V. J. Fortuna, <a href="http://www.bibliotecadigital.unicamp.br/document/?code=vtls000366716">Distancias de Transposito entre Genomas</a>, Master's Thesis, Universidade Estadual de Campinas, 2005. [_Anthony Labarre_, Feb 16 2009]
%F a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k).
%F a(n) = hypergeom([1, -n/2, -n/2+1/2], [], 2).
%F Conjecture: 2*a(n) -2*a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Aug 19 2014
%F a(n) ~ n! * (exp(sqrt(2)) + (-1)^n * exp(-sqrt(2))) / 2^(n/2+1). - _Vaclav Kotesovec_, Mar 20 2015
%F From _Vladimir Reshetnikov_, Oct 27 2015: (Start)
%F a(n) = ((-1)^n * exp(-sqrt(2)) * Gamma(n+2,-sqrt(2)) + exp(sqrt(2)) * Gamma(n+2,sqrt(2))) / ((n+1) * 2^(n/2+1)).
%F For even n, a(n) ~ 2^(1/2-n/2)*exp(-n)*n^(n+1/2)*sqrt(Pi)*cosh(sqrt(2)).
%F For odd n, a(n) ~ 2^(1/2-n/2)*exp(-n)*n^(n+1/2)*sqrt(Pi)*sinh(sqrt(2)). (End)
%e G.f. = 1 + x + 2*x^2 + 4*x^3 + 13*x^4 + 41*x^5 + 196*x^6 + 862*x^7 + ...
%p a := n -> ((-1)^n*exp(-sqrt(2))*GAMMA(n+2,-sqrt(2))+exp(sqrt(2))*GAMMA(n+2, sqrt(2)))/((n+1)*2^(n/2+1)): seq(simplify(a(n)),n=0..23); # after V. Reshetnikov, _Peter Luschny_, Oct 27 2015
%t CoefficientList[Series[E^x/(1-x^2/2), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Mar 20 2015 *)
%t Table[HypergeometricPFQ[{1, -n/2, -n/2 + 1/2}, {}, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 27 2015 *)
%o (PARI) nmax=200; \\ _Stanislav Sykora_, Nov 03 2016
%o a=vector(nmax,m,n=m-1,sum(k=0,n\2,n!/(2^k*(n-2*k)!)))
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp(x + x * O(x^n)) / (1 - x^2/2), n))}; /* _Michael Somos_, Nov 03 2016 */
%Y Cf. A126725, A157018.
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Oct 19 2003
%E Name corrected by _Vaclav Kotesovec_, Mar 20 2015
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