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A000756 Boustrophedon transform of sequence 1,1,0,0,0,0,... 1
1, 2, 3, 5, 13, 41, 157, 699, 3561, 20401, 129881, 909523, 6948269, 57504201, 512516565, 4894172027, 49851629137, 539521049441, 6182455849009, 74781598946211, 952148890494165, 12729293006112121, 178281831561868013 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

S. N. Gladkovskii, Continued fraction expansion for function sec(x) + tan(x),  arXiv:1208.2243v1 [math.HO], Aug 09 2012

S. N. Gladkovskii, On the continued fraction expansion for functions 1/sin(x) + cot(x) and sec(x) + tan(x), Nov 12 2012

Peter Luschny, An old operation on sequences: the Seidel transform

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

N. J. A. Sloane, Transforms

Wikipedia, Boustrophedon_transform

Index entries for sequences related to boustrophedon transform

FORMULA

E.g.f.: (1+x)*( tan(x) + sec(x) ).

From Sergei N. Gladkovskii Dec 03 2012 (Start)

E.g.f.: (1+x)*( 1 + x/U(0) ); U(k)= 4k + 1 - x/(2-x/(4k + 3 + x/(2+x/U(k+1) )));(continued fraction, , 4-step).

E.g.f.: (1+x)*(1 + 2*x/(U(0) - x) ) where U(k)= 4*k + 2 - x^2/U(k+1);(continued fraction, 1-step).

(End)

a(n) ~ n! * (Pi+2)*(2/Pi)^(n+1). - Vaclav Kotesovec, Oct 02 2013

For n > 0: a(n) = A000111(n) + n*A000111(n-1). - Reinhard Zumkeller, Nov 03 2013

MATHEMATICA

CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

t[n_, 0] := If[n < 2, 1, 0]; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

PROG

(Sage) Algorithm of L. Seidel (1877)

def A000756_list(n) :

    R = []; A = {-1:1, 0:1}; k = 0; e = 1

    for i in (0..n) :

        Am = 0; A[k + e] = 0; e = -e

        for j in (0..i) : Am += A[k]; A[k] = Am; k += e

        R.append(A[-i//2] if i%2 == 0 else A[i//2])

    return R

A000756_list(22) # Peter Luschny, May 27 2012

(PARI)

x='x+O('x^66);

Vec(serlaplace((1+x)*(tan(x)+ 1/cos(x))))

/* Joerg Arndt, May 28 2012 */

(Haskell)

a000756 n = sum $ zipWith (*) (a109449_row n) (1 : 1 : [0, 0 ..])

-- Reinhard Zumkeller, Nov 03 2013

CROSSREFS

Cf. A109449.

Sequence in context: A038560 A240838 A238814 * A192241 A093999 A042445

Adjacent sequences:  A000753 A000754 A000755 * A000757 A000758 A000759

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 20 15:54 EST 2018. Contains 299380 sequences. (Running on oeis4.)