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A008280
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Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows.
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13
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,9
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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T(n,m) = abs( Sum_{k=0..n} C(m,k)*Euler(n-m+k) ). - Vladimir Kruchinin, Apr 06 2015
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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