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A000708 a(n) = E(n+1)-2E(n), where E(i) is the Euler number A000111(i).
(Formerly M4188 N1745)
5
-1, -1, 0, 1, 6, 29, 150, 841, 5166, 34649, 252750, 1995181, 16962726, 154624469, 1505035350, 15583997521, 171082318686, 1985148989489, 24279125761950, 312193418011861, 4210755676649046, 59445878286889709, 876726137720576550 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) mod 10 for n>=2 is the periodic sequence repeat: 0, 1, 6, 9.

For n >= 2 a(n) is the number of permutations on [n] that have n-2 "sequences" (which are maximal monotone runs in Comtet terminology) and start increasing. - Michael Somos, Aug 28 2013

REFERENCES

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

John Cerkan, Table of n, a(n) for n = 0..482

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.

E. Netto, Lehrbuch der Combinatorik, Annotated scanned copy of pages 112-113 only.

Eric Weisstein's MathWorld, Polylogarithm.

FORMULA

E.g.f.: (1 - 2*cos(x)) / (1 - sin(x)).

a(n) ~ n! * 2*n*(2/Pi)^(n+2). - Vaclav Kotesovec, Oct 08 2013

a(n) = 2*abs(PolyLog(-n-1, i)) - 4*abs(PolyLog(-n, i)) for n>0, with a(0) = -1. - Jean-François Alcover, Jul 02 2017

EXAMPLE

G.f. = -1 - x + x^3 + 6*x^4 + 29*x^5 + 150*x^6 + 841*x^7 + 5166*x^8 + 34649*x^9 + ...

a(3) = 1 with permutation 123. a(4) = 6 with permutations 1243, 1342, 1432, 2341, 2431, 3421.

MAPLE

seq(i! * coeff(series((1 + (tan(t) + sec(t))^2 - 4*(tan(t) + sec(t))) / 2, t, 35), t, i), i=2..24); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

MATHEMATICA

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - 2 Cos[x]) / (1 - Sin[x]), {x, 0, n}]; (* Michael Somos, Aug 28 2013 *)

nmax = 22; ee = Table[2^n*EulerE[n, 1] + EulerE[n], {n, 0, nmax+1}]; dd = Table[Differences[ee, n][[1]] // Abs, {n, 0, nmax+1}]; a[n_] := dd[[n+2]] - 2dd[[n+1]]; a[0] = -1; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Feb 10 2016, after Paul Curtz *)

Table[If[n == 0, -1, 2 Abs[PolyLog[-n-1, I]] - 4 Abs[PolyLog[-n, I]]], {n, 0, 22}] (* Jean-François Alcover, Jul 01 2017 *)

PROG

(PARI) x='x+O('x^99); Vec(serlaplace((1-2*cos(x))/(1-sin(x))))

(Python)

from mpmath import polylog, j, mp

mp.dps=20

def a(n): return -1 if n==0 else int(2*abs(polylog(-n - 1, j)) - 4*abs(polylog(-n, j)))

print [a(n) for n in xrange(23)] # Indranil Ghosh, Jul 02 2017

CROSSREFS

Apart from initial terms, equals (1/2)*A001758. A diagonal of A008970.

Sequence in context: A292034 A108982 A059724 * A027248 A192481 A020090

Adjacent sequences:  A000705 A000706 A000707 * A000709 A000710 A000711

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

Corrected and extended by T. D. Noe, Oct 25 2006

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

STATUS

approved

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Last modified October 18 10:03 EDT 2018. Contains 316320 sequences. (Running on oeis4.)