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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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14
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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;
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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
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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.
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.
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LINKS
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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).
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FORMULA
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EXAMPLE
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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]
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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.
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)])
(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
(Python) # Python 3.2 or higher required.
from itertools import accumulate
for n in range(10):
blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(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
(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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KEYWORD
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AUTHOR
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STATUS
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approved
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