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A008280 Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows. 14

%I

%S 1,0,1,1,1,0,0,1,2,2,5,5,4,2,0,0,5,10,14,16,16,61,61,56,46,32,16,0,0,

%T 61,122,178,224,256,272,272,1385,1385,1324,1202,1024,800,544,272,0,0,

%U 1385,2770,4094,5296,6320,7120,7664,7936,7936

%N Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows.

%C The earliest known reference for this triangle is Seidel (1877). - _Don Knuth_, Jul 13 2007

%C Sum of row n = A000111(n+1). - _Reinhard Zumkeller_, Nov 01 2013

%D M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 110.

%D A. J. Kempner, On the shape of polynomial curves, Tohoku Math. J., 37 (1933), 347-362.

%D A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 53.

%D R. P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 141, Cambridge University Press (2012), p. 128, 174, 175.

%H Vincenzo Librandi, <a href="/A008280/b008280.txt">Table of n, a(n) for n = 0..1000</a>

%H V. I. Arnold, <a href="http://dx.doi.org/10.1215/S0012-7094-91-06323-4">Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics</a>, Duke Math. J. 63 (1991), 537-555.

%H V. I. Arnold, <a href="http://mi.mathnet.ru/eng/umn4470">The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups</a>, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.

%H M. D. Atkinson, <a href="http://dx.doi.org/10.1016/0020-0190(85)90057-2">Zigzag permutations and comparisons of adjacent elements</a>, Information Processing Letters 21 (1985), 187-189.

%H Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.

%H B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%H J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).

%H C. Poupard, <a href="http://dx.doi.org/10.1016/0012-365X(82)90293-X">De nouvelles significations énumératives des nombres d'Entringer</a>, Discrete Math., 38 (1982), 265-271.

%H Sanjay Ramassamy, <a href="https://arxiv.org/abs/1712.08666">Modular periodicity of the Euler numbers and a sequence by Arnold</a>, arXiv:1712.08666 [math.CO], 2017.

%H L. Seidel, <a href="http://publikationen.badw.de/de/003384831/pdf/CC%20BY">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187; see Beilage 5, pp. 183-184.

%H R. Street, <a href="https://arxiv.org/abs/math/0303267">Trees, permutations and the tangent function</a>, arXiv:math/0303267 [math.HO], 2003.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>

%H <a href="/index/Bo#boustrophedon"> Index entries for sequences related to boustrophedon transform</a>

%F T(n,m) = abs( Sum_{k=0..n} C(m,k)*Euler(n-m+k) ). - _Vladimir Kruchinin_, Apr 06 2015

%e This version of the triangle begins:

%e 1

%e 0 1

%e 1 1 0

%e 0 1 2 2

%e 5 5 4 2 0

%e 0 5 10 14 16 16

%e See A008281 and A108040 for other versions.

%t max = 9; t[0, 0] = 1; t[n_, m_] /; n < m || m < 0 = 0; t[n_, m_] := t[n, m] = Sum[t[n-1, n-k], {k, m}]; tri = Table[t[n, m], {n, 0, max}, {m, 0, n}]; Flatten[ {Reverse[#[[1]]], #[[2]]} & /@ Partition[tri, 2]] (* _Jean-François Alcover_, Oct 24 2011 *)

%o (Sage) # Algorithm of L. Seidel (1877)

%o # Prints the first n rows of the triangle.

%o def A008280_triangle(n) :

%o A = {-1:0, 0:1}

%o k = 0; e = 1

%o for i in range(n) :

%o Am = 0

%o A[k + e] = 0

%o e = -e

%o for j in (0..i) :

%o Am += A[k]

%o A[k] = Am

%o k += e

%o print([A[z] for z in (-i//2..i//2)])

%o A008280_triangle(10) # _Peter Luschny_, Jun 02 2012

%o (Haskell)

%o a008280 n k = a008280_tabl !! n !! k

%o a008280_row n = a008280_tabl !! n

%o a008280_tabl = ox True a008281_tabl where

%o ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss

%o -- _Reinhard Zumkeller_, Nov 01 2013

%o (Python) # Python 3.2 or higher required.

%o from itertools import accumulate

%o A008280_list = blist = [1]

%o for n in range(10):

%o blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))

%o A008280_list.extend(blist)

%o print(A008280_list) # Chai Wah Wu, Sep 20 2014

%o (Maxima)

%o T(n, m):=abs(sum(binomial(m, k)*euler(n-m+k), k, 0, m)); /* _Vladimir Kruchinin_, Apr 06 2015 */

%Y Cf. A008281, A108040, A058257.

%Y Cf. A000657 (central terms); A227862.

%K nonn,tabl,nice

%O 0,9

%A _N. J. A. Sloane_

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Last modified August 3 08:21 EDT 2021. Contains 346435 sequences. (Running on oeis4.)