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

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, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice

Comments

The earliest known reference for this triangle is Seidel (1877). - Don Knuth, Jul 13 2007

Sum of row n = A000111(n+1). - Reinhard Zumkeller, Nov 01 2013

Examples

			This version of the triangle begins:
[0] [   1]
[1] [   0,    1]
[2] [   1,    1,    0]
[3] [   0,    1,    2,    2]
[4] [   5,    5,    4,    2,    0]
[5] [   0,    5,   10,   14,   16,   16]
[6] [  61,   61,   56,   46,   32,   16,    0]
[7] [   0,   61,  122,  178,  224,  256,  272,  272]
[8] [1385, 1385, 1324, 1202, 1024,  800,  544,  272,    0]
[9] [   0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]
See A008281 and A108040 for other versions.
		

References

• 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.
• J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 110.
• A. J. Kempner, On the shape of polynomial curves, Tohoku Math. J., 37 (1933), 347-362.
• A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 53.
• R. P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 141, Cambridge University Press (2012), p. 128, 174, 175.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• V. I. Arnold, Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537-555.
• V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
• M. D. Atkinson, Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
• Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
• Dominique Foata, Guo-Niu Han, and Volker Strehl, The Entringer-Poupard matrix sequence. Linear Algebra Appl. 512, 71-96 (2017). Example 4.3.
• Ira M. Gessel, Counting up-up-or-down-down permutations, arXiv:2411.16113 [math.CO], 2024.
• Boris Gourévitch, L'univers de Pi
• 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 44-54 1996 (Abstract, pdf, ps).
• Christiane Poupard, De nouvelles significations énumératives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
• Sanjay Ramassamy, Modular periodicity of the Euler numbers and a sequence by Arnold, arXiv:1712.08666 [math.CO], 2017.
• L. 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; see Beilage 5, pp. 183-184.
• Ross Street, Trees, permutations and the tangent function, arXiv:math/0303267 [math.HO], 2003.
• Wikipedia, Boustrophedon transform
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A008281, A108040, A058257.

Cf. A000657 (central terms); A227862.

Programs

Haskell

a008280 n k = a008280_tabl !! n !! k
a008280_row n = a008280_tabl !! n
a008280_tabl = ox True a008281_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
-- Reinhard Zumkeller, Nov 01 2013

Mathematica

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[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 *)

Maxima

T(n, m):=abs(sum(binomial(m, k)*euler(n-m+k), k, 0, m)); /* Vladimir Kruchinin, Apr 06 2015 */

Python

# Python 3.2 or higher required.
from itertools import accumulate
A008280_list = blist = [1]
for n in range(10):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    A008280_list.extend(blist)
print(A008280_list) # Chai Wah Wu, Sep 20 2014

Python

# Uses function seidel from A008281.
def A008280row(n): return seidel(n) if n % 2 else seidel(n)[::-1]
for n in range(8): print(A008280row(n)) # Peter Luschny, Jun 01 2022

Sage

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A008280_triangle(n) :
    A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        print([A[z] for z in (-i//2..i//2)])
A008280_triangle(10) # Peter Luschny, Jun 02 2012

Formula

T(n,m) = abs( Sum_{k=0..n} C(m,k)*Euler(n-m+k) ). - Vladimir Kruchinin, Apr 06 2015

E.g.f.: (cos(x) + sin(x))/cos(x+y). - Ira M. Gessel, Nov 18 2024