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A008281 Triangle of Euler-Bernoulli or Entringer numbers read by rows. 11
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; text; internal format)
OFFSET

0,9

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.

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.

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

LINKS

Vincenzo Librandi and Alois P. Heinz, Rows n = 0..140 (rows n = 0..43 from Vincenzo Librandi)

B. Gourevitch, L'univers de Pi

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, Arxiv preprint arXiv:1110.5272, 2011

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).

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).

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.

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( procname(h, h-1)) ;

    else

        RETURN( add(procname(h-1, j), j=h-k..h-1) ) ;

    fi ;

end: # - R. J. Mathar, Nov 27 2006

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}]] (* Jean-Fran├žois Alcover, May 31 2011, after formula *)

PROG

(Haskell)

a008281 n k = a008281_tabl !! n !! k

a008281_row n = a008281_tabl !! n

a008281_tabl = iterate (scanl (+) 0 . reverse) [1]

-- Reinhard Zumkeller, Sep 10 2013

CROSSREFS

Cf. A008280.

Sequence in context: A159916 A159286 A006462 * A094671 A202015 A193350

Adjacent sequences:  A008278 A008279 A008280 * A008282 A008283 A008284

KEYWORD

nonn,tabl,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 17 08:58 EDT 2014. Contains 240634 sequences.