A000756 Boustrophedon transform of sequence 1,1,0,0,0,0,...

1, 2, 3, 5, 13, 41, 157, 699, 3561, 20401, 129881, 909523, 6948269, 57504201, 512516565, 4894172027, 49851629137, 539521049441, 6182455849009, 74781598946211, 952148890494165, 12729293006112121, 178281831561868013

Offset

0,2

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], 2012.

S. N. Gladkovskii, On the continued fraction expansion for functions 1/sin(x) + cot(x) and sec(x) + tan(x), 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 transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).

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. [Access through ZOBODAT]

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) = 4*k + 1 - x/(2 - x/(4*k + 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
(Python)
from itertools import islice, accumulate
def A000756_gen(): # generator of terms
    yield from (1, 2)
    blist = (1, 2)
    while True:
        yield (blist:=tuple(accumulate(reversed(blist), initial=0)))[-1]
A000756_list = list(islice(A000756_gen(), 40)) # Chai Wah Wu, Jun 09-11 2022

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