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A001758 Number of quasi-alternating permutations of length n.
(Formerly M2027 N0800)
5
0, 2, 12, 58, 300, 1682, 10332, 69298, 505500, 3990362, 33925452, 309248938, 3010070700, 31167995042, 342164637372, 3970297978978, 48558251523900, 624386836023722, 8421511353298092, 118891756573779418, 1753452275441153100, 26967372781086764402 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

The number of permutations of [n] with n-2 sequences (see Comtet).

REFERENCES

D. Andre, Etude sur les maxima, minima et sequences des permutations, Annales scientifiques de l'E.N.S. 3e serie, tome 1 (1884), 121-134.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 113.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 2..100

M. E. Estanave, Sur les coefficients des développements en séries de tang x, séc x et d'autres fonctions. Caractères de périodicité que présentent les chiffres des unités de ces coefficients, Bulletin de la S.M.F., 30 (1902), pp. 220-226. See p. 223.

FORMULA

E.g.f.: 3+2*x + u(x)^2-4*u(x) where u(x)=(tan(x)+sec(x)). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

E.g.f.: 2 * (1 + x + (1 - 2*cos(x)) / (1 - sin(x))). - Michael Somos, Aug 28 2012

Asymptotics: a(n) ~ 8(2/Pi)^(n+1)((n+1)/Pi-1))n!

a(n) = A001250(n+1) - 2*A001250(n). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

EXAMPLE

G.f. = 2*x^3 + 12*x^4 + 58*x^5 + 300*x^6 + 1682*x^7 + 10332*x^8 + 69298*x^9 + ...

MAPLE

seq(i!*coeff(series((tan(t)+sec(t))^2-4*(tan(t)+sec(t)), t, 35), t, i), i=2..24); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

MATHEMATICA

With[{nn=30}, Join[{1}, Drop[CoefficientList[Series[(Tan[x]+Sec[x])^2- 4(Tan[x]+Sec[x]), {x, 0, nn}], x] Range[0, nn]!, 3]]] (* Harvey P. Dale, Oct 01 2011 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (u (u - 4) /. u -> Tan[x] + Sec[x]) + 3 + 2 x, {x, 0, n}]]; (* Michael Somos, Oct 24 2015 *)

Table[4 Abs[PolyLog[-n-1, I]] - 8 Abs[PolyLog[-n, I]], {n, 2, 23}] (* Jean-François Alcover, Jul 01 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); 2 * n! * polcoeff( 1 + x + (1 - 2 * cos(x + A)) / (1 - sin(x + A)), n))}; /* Michael Somos, Aug 28 2012 */

(PARI) x='x+O('x^99); concat(0, Vec(serlaplace(2*(1+x+(1-2*cos(x))/(1-sin(x)))))) \\ Altug Alkan, Jul 01 2017

CROSSREFS

Essentially the same as 2*A000708.

The diagonal P(n, n-2) of A059427.

Cf. A001759, A001760, A001250.

See A008970 for formulas.

Sequence in context: A054145 A285364 A282435 * A037133 A009618 A143770

Adjacent sequences:  A001755 A001756 A001757 * A001759 A001760 A001761

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001

Edited by N. J. A. Sloane, Aug 27 2012

STATUS

approved

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Last modified October 17 14:25 EDT 2018. Contains 316281 sequences. (Running on oeis4.)