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A087214
Expansion of e.g.f.: exp(x)/(1-x^2/2).
7
1, 1, 2, 4, 13, 41, 196, 862, 5489, 31033, 247006, 1706816, 16302397, 133131649, 1483518128, 13978823146, 178022175361, 1901119947857, 27237392830234, 325091511083548, 5175104637744461, 68269217327545081, 1195449171318970492, 17272111983868905494
OFFSET
0,3
COMMENTS
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
From Stanislav Sykora, Nov 03 2016: (Start)
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.
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.
(End)
LINKS
Zanoni Dias and Joao Meidanis, Sorting by Prefix Transpositions, 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]
Zanoni Dias, Vinicius Fortuna and Joao Meidanis, Sorting by Prefix Transpositions, 2004.
V. J. Fortuna, Distancias de Transposito entre Genomas, Master's Thesis, Universidade Estadual de Campinas, 2005. [Anthony Labarre, Feb 16 2009]
FORMULA
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k).
a(n) = hypergeom([1, -n/2, -n/2+1/2], [], 2).
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
a(n) ~ n! * (exp(sqrt(2)) + (-1)^n * exp(-sqrt(2))) / 2^(n/2+1). - Vaclav Kotesovec, Mar 20 2015
From Vladimir Reshetnikov, Oct 27 2015: (Start)
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)).
For even n, a(n) ~ 2^(1/2-n/2)*exp(-n)*n^(n+1/2)*sqrt(Pi)*cosh(sqrt(2)).
For odd n, a(n) ~ 2^(1/2-n/2)*exp(-n)*n^(n+1/2)*sqrt(Pi)*sinh(sqrt(2)). (End)
EXAMPLE
G.f. = 1 + x + 2*x^2 + 4*x^3 + 13*x^4 + 41*x^5 + 196*x^6 + 862*x^7 + ...
MAPLE
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
MATHEMATICA
CoefficientList[Series[E^x/(1-x^2/2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 20 2015 *)
Table[HypergeometricPFQ[{1, -n/2, -n/2 + 1/2}, {}, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 27 2015 *)
PROG
(PARI) nmax=200; \\ Stanislav Sykora, Nov 03 2016
a=vector(nmax, m, n=m-1, sum(k=0, n\2, n!/(2^k*(n-2*k)!)))
(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 */
CROSSREFS
Sequence in context: A325578 A118930 A355194 * A259239 A243107 A002771
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Oct 19 2003
EXTENSIONS
Name corrected by Vaclav Kotesovec, Mar 20 2015
STATUS
approved