A087214 Expansion of e.g.f.: exp(x)/(1-x^2/2).

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

Stanislav Sykora, Table of n, a(n) for n = 0..199

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

Cf. A126725, A157018.

Sequence in context: A325578 A118930 A355194 * A259239 A243107 A002771

Adjacent sequences: A087211 A087212 A087213 * A087215 A087216 A087217

Keyword

nonn

Author

Vladeta Jovovic, Oct 19 2003

Extensions

Name corrected by Vaclav Kotesovec, Mar 20 2015

Status

approved