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Search: id:A008282
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| A008282 |
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Triangle of Euler-Bernoulli or Entringer numbers read by rows: T(n,k) is the number of down-up permutations of n+1 starting with k+1. |
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13
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| 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 10, 14, 16, 16, 16, 32, 46, 56, 61, 61, 61, 122, 178, 224, 256, 272, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936, 7936, 15872
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Triangle begins
1
1 1
1 2 2
2 4 5 5
5 10 14 16 16
16 32 46 56 61 61
...
Each row is constructed by forming the partial sums of the previous row, reading from the right, and repeating the final term.
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REFERENCES
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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.
R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
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LINKS
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J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
B. Gourevitch, L'univers de Pi
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).
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FORMULA
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T(n, k)=sum((-1)^i*binomial(k, 2i+1)*E[n-2i-1], i=0..floor((k-1)/2))= sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (k<n), T(n, n)=E[n]. T(n, n)=E[n]; T(n, k)=sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (k<n), T(n, n)=E[n]. where E(j)=A000111(j)=j!*[x^j]((sec(x)+tan(x)) are the up/down or Euler numbers. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2004
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EXAMPLE
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T(4,3)=5 because we have 41325,41523,42314,42513 and 43512.
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MAPLE
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f:=series(sec(x)+tan(x), x=0, 25): E[0]:=1: for n from 1 to 20 do E[n]:=n!*coeff(f, x^n) od: T:=proc(n, k) if k<n then sum((-1)^i*binomial(k, 2*i+1)*E[n-2*i-1], i=0..floor((k-1)/2)) elif k=n then E[n] else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..10);
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CROSSREFS
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Cf. A010094, A000111, A099959, A009766.
Sequence in context: A035002 A032578 A035659 this_sequence A074765 A029045 A057591
Adjacent sequences: A008279 A008280 A008281 this_sequence A008283 A008284 A008285
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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njas
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