0,9

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

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.

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.

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

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.

B. Gourevitch, 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 on transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

R. Street, Trees, permutations and the tangent function

Wikipedia, Boustrophedon_transform

Index entries for sequences related to boustrophedon transform

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

This version of the triangle begins:

.............1

...........0...1

.........1...1...0

.......0...1...2...2

.....5...5...4...2...0

...0...5..10..14..16..16

See A008281 and A108040 for other versions.

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 *)

(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

(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

(Python)

# Python 3.2 or higher required.

from itertools import accumulate

A008280_list = blist = [1]

for n in range(30):

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

....A008280_list.extend(blist) # Chai Wah Wu, Sep 20 2014

(Maxima)

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

Cf. A008281, A108040, A058257.

Cf. A000657 (central terms); A227862.

Sequence in context: A257943 A236935 * A239005 A213187 A195710 A200997

Adjacent sequences:  A008277 A008278 A008279 * A008281 A008282 A008283

nonn,tabl,nice

N. J. A. Sloane

approved