cp's OEIS Frontend

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

%I A008280 #103 Sep 03 2025 05:02:36
%S A008280 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 A008280 61,122,178,224,256,272,272,1385,1385,1324,1202,1024,800,544,272,0,0,
%U A008280 1385,2770,4094,5296,6320,7120,7664,7936,7936
%N A008280 Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows.
%C A008280 The earliest known reference for this triangle is Seidel (1877). - _Don Knuth_, Jul 13 2007
%C A008280 Sum of row n = A000111(n+1). - _Reinhard Zumkeller_, Nov 01 2013
%D A008280 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 A008280 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 110.
%D A008280 A. J. Kempner, On the shape of polynomial curves, Tohoku Math. J., 37 (1933), 347-362.
%D A008280 A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 53.
%D A008280 R. P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 141, Cambridge University Press (2012), p. 128, 174, 175.
%H A008280 Vincenzo Librandi, <a href="/A008280/b008280.txt">Table of n, a(n) for n = 0..1000</a>
%H A008280 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 A008280 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 A008280 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 A008280 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 A008280 Dominique Foata, Guo-Niu Han, and Volker Strehl, <a href="https://doi.org/10.1016/j.laa.2016.09.016">The Entringer-Poupard matrix sequence</a>. Linear Algebra Appl. 512, 71-96 (2017). Example 4.3.
%H A008280 Ira M. Gessel, <a href="https://arxiv.org/abs/2411.16113">Counting up-up-or-down-down permutations</a>, arXiv:2411.16113 [math.CO], 2024.
%H A008280 Boris Gourévitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H A008280 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A008280 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 A008280 Christiane 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 A008280 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 A008280 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 A008280 Ross Street, <a href="https://arxiv.org/abs/math/0303267">Trees, permutations and the tangent function</a>, arXiv:math/0303267 [math.HO], 2003.
%H A008280 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H A008280 <a href="/index/Bo#boustrophedon"> Index entries for sequences related to boustrophedon transform</a>
%F A008280 T(n,m) = abs( Sum_{k=0..n} C(m,k)*Euler(n-m+k) ). - _Vladimir Kruchinin_, Apr 06 2015
%F A008280 E.g.f.: (cos(x) + sin(x))/cos(x+y). - _Ira M. Gessel_, Nov 18 2024
%e A008280 This version of the triangle begins:
%e A008280   [0] [   1]
%e A008280   [1] [   0,    1]
%e A008280   [2] [   1,    1,    0]
%e A008280   [3] [   0,    1,    2,    2]
%e A008280   [4] [   5,    5,    4,    2,    0]
%e A008280   [5] [   0,    5,   10,   14,   16,   16]
%e A008280   [6] [  61,   61,   56,   46,   32,   16,    0]
%e A008280   [7] [   0,   61,  122,  178,  224,  256,  272,  272]
%e A008280   [8] [1385, 1385, 1324, 1202, 1024,  800,  544,  272,    0]
%e A008280   [9] [   0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]
%e A008280 See A008281 and A108040 for other versions.
%t A008280 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 *)
%t A008280 T[0,0] := 1; T[n_?OddQ,k_]/;0<=k<=n := T[n,k]=T[n,k-1]+T[n-1,k-1]; T[n_?EvenQ,k_]/;0<= k<=n := T[n,k]=T[n,k+1]+T[n-1,k]; T[n_,k_] := 0; Flatten@Table[T[n,k], {n,0,9}, {k,0,n}] (* _Oliver Seipel_, Nov 24 2024 *)
%o A008280 (Sage) # Algorithm of L. Seidel (1877)
%o A008280 # Prints the first n rows of the triangle.
%o A008280 def A008280_triangle(n) :
%o A008280     A = {-1:0, 0:1}
%o A008280     k = 0; e = 1
%o A008280     for i in range(n) :
%o A008280         Am = 0
%o A008280         A[k + e] = 0
%o A008280         e = -e
%o A008280         for j in (0..i) :
%o A008280             Am += A[k]
%o A008280             A[k] = Am
%o A008280             k += e
%o A008280         print([A[z] for z in (-i//2..i//2)])
%o A008280 A008280_triangle(10) # _Peter Luschny_, Jun 02 2012
%o A008280 (Haskell)
%o A008280 a008280 n k = a008280_tabl !! n !! k
%o A008280 a008280_row n = a008280_tabl !! n
%o A008280 a008280_tabl = ox True a008281_tabl where
%o A008280   ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
%o A008280 -- _Reinhard Zumkeller_, Nov 01 2013
%o A008280 (Python) # Python 3.2 or higher required.
%o A008280 from itertools import accumulate
%o A008280 A008280_list = blist = [1]
%o A008280 for n in range(10):
%o A008280     blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
%o A008280     A008280_list.extend(blist)
%o A008280 print(A008280_list) # _Chai Wah Wu_, Sep 20 2014
%o A008280 (Python) # Uses function seidel from A008281.
%o A008280 def A008280row(n): return seidel(n) if n % 2 else seidel(n)[::-1]
%o A008280 for n in range(8): print(A008280row(n)) # _Peter Luschny_, Jun 01 2022
%o A008280 (Maxima)
%o A008280 T(n, m):=abs(sum(binomial(m, k)*euler(n-m+k), k, 0, m)); /* _Vladimir Kruchinin_, Apr 06 2015 */
%Y A008280 Cf. A008281, A108040, A058257.
%Y A008280 Cf. A000657 (central terms); A227862.
%K A008280 nonn,tabl,nice,changed
%O A008280 0,9
%A A008280 _N. J. A. Sloane_