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A001758
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Number of quasi-alternating permutations of length n.
(Formerly M2027 N0800)
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5
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0, 2, 12, 58, 300, 1682, 10332, 69298, 505500, 3990362, 33925452, 309248938, 3010070700, 31167995042, 342164637372, 3970297978978, 48558251523900, 624386836023722, 8421511353298092, 118891756573779418
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OFFSET
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2,2
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COMMENTS
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The number of permutations of [n] with n-2 sequences (see Comtet).
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..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.
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FORMULA
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E.g.f.: 3+2*x + u(x)^2-4*u(x) where u(x)=(tan(x)+sec(x)).
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).
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EXAMPLE
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2*x^3 + 12*x^4 + 58*x^5 + 300*x^6 + 1682*x^7 + 10332*x^8 + 69298*x^9 + ...
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MAPLE
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seq(i!*coeff(series((tan(t)+sec(t))^2-4*(tan(t)+sec(t)), t, 35), t, i), i=2..24);
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = local(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 */
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CROSSREFS
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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: A094780 A100103 A054145 * A037133 A009618 A143770
Adjacent sequences: A001755 A001756 A001757 * A001759 A001760 A001761
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001
E.g.f., asymptotics and Maple code from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001
Edited by N. J. A. Sloane, Aug 27 2012
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STATUS
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approved
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