The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A008281 Triangle of Euler-Bernoulli or Entringer numbers read by rows. 13
 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. The sequence (a(n,n)) of the last element of each row is that of Euler up/down numbers A000111, the Boustrophedon transform of sequence A000007 = (0^n) = (1, 0, 0, 0, ...). - M. F. Hasler, Oct 07 2017 REFERENCES 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), Birkhäuser, 1999 Basel, p. 31. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997. LINKS Alois P. Heinz, Rows n = 0..140 (rows n = 0..43 from Vincenzo Librandi) 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. B. Gourevitch, L'univers de Pi M. Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011. J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps). C. Poupard, De nouvelles significations énumératives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271. Heesung Shin and Jiang Zeng, More bijections for Entringer and Arnold families, arXiv:2006.00507 [math.CO], 2020. OEIS Wiki, Boustrophedon transform. FORMULA a(0,0)=1, a(n,m)=0 if n < m, a(n,m)=0 if m < 0, otherwise a(n,m) = Sum_{k=1..m} a(n-1,n-k). T(n, k) = T(n, k-1) + T(n-1, n-k) for k > 0, T(n, 0) = 0^n. - Peter Luschny, Sep 30 2023 EXAMPLE This version of the triangle begins: [0] [1] [1] [0, 1] [2] [0, 1, 1] [3] [0, 1, 2, 2] [4] [0, 2, 4, 5, 5] [5] [0, 5, 10, 14, 16, 16] [6] [0, 16, 32, 46, 56, 61, 61] [7] [0, 61, 122, 178, 224, 256, 272, 272] [8] [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] [9] [0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936] 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 # Alternative: T := proc(n, k) option remember; ifelse(k=0, 0^n, T(n, k-1) + T(n-1, n-k)) end: # Peter Luschny, Sep 30 2023 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-François 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 (Python) # Python 3.2 or higher required. from itertools import accumulate A008281_list = blist = [1] for _ in range(30): blist = [0]+list(accumulate(reversed(blist))) A008281_list.extend(blist) # Chai Wah Wu, Sep 18 2014 (Python) from functools import cache @cache def seidel(n): if n == 0: return [1] rowA = seidel(n - 1) row = [0] + seidel(n - 1) row[1] = row[n] for k in range(2, n + 1): row[k] = row[k - 1] + rowA[n - k] return row def A008281row(n): return seidel(n) for n in range(8): print(A008281row(n)) # Peter Luschny, Jun 01 2022 CROSSREFS Cf. A008280, A000111, A108040. Sequence in context: A261277 A006462 A278483 * A094671 A354826 A202015 Adjacent sequences: A008278 A008279 A008280 * A008282 A008283 A008284 KEYWORD nonn,tabl,nice,easy AUTHOR N. J. A. Sloane STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 29 10:24 EDT 2024. Contains 372938 sequences. (Running on oeis4.)