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A008281
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Triangle of Euler-Bernoulli or Entringer numbers read by rows.
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9
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1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 4, 5, 5, 0, 5, 10, 14, 16, 16, 0, 16, 32, 46, 56, 61, 61, 0, 61, 122, 178, 224, 256, 272, 272, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| Zig-Zag numbers (see the Conway and Guy reference p. 110 and the J.-P. Delahaye reference, p. 31).
Approximation to Pi: 2*n*a(n-1,n-1)/a(n,n), n>=3. See A132049(n)/A132050(n). See the Delahaye reference, p. 31.
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REFERENCES
| Arnold, V. I., 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.
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, p. 110.
J.-P. Delahaye, Pi - die Story (German translation), Birkhaeuser, 1999 Basel, p. 31. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, Arxiv preprint arXiv:1110.5272, 2011
C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
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LINKS
| 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
| a(0,0)=1, a(n,m)=0 if n<m, a(n,m)=0 if m<0 else sum(a(n-1,n-k),k=1..m).
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EXAMPLE
| This version of the triangle begins:
.............1
...........0...1
.........0...1...1
.......0...1...2...2
.....0...2...4...5...5
...0...5..10..14..16..16
See A008280 and A108040 for other versions.
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MAPLE
| A008281 := proc(h, k) option remember ; if h=1 and k=1 or h=0 then RETURN(1) ; elif h>=1 and k> h then RETURN(0) ; elif h=k then RETURN( A008281(h, h-1)) ; else RETURN( add(A008281(h-1, j), j=h-k..h-1) ) ; fi ; end: for h from 0 to 10 do for k from 0 to h do printf("%d, ", A008281(h, k)) ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 27 2006
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MATHEMATICA
| a[0, 0] = 1; a[n_, m_] /; (n < m || m < 0) = 0; a[n_, m_] := a[n, m] = Sum[a[n-1, n-k], {k, m}]; Flatten[Table[a[n, m], {n, 0, 9}, {m, 0, n}]] (* From Jean-François Alcover, May 31 2011, after formula *)
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CROSSREFS
| Cf. A008280.
Sequence in context: A159916 A159286 A006462 * A094671 A202015 A193350
Adjacent sequences: A008278 A008279 A008280 * A008282 A008283 A008284
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KEYWORD
| nonn,tabl,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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