A008281 Triangle of Euler-Bernoulli or Entringer numbers read by rows.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

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

Examples

			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.
		

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.

Crossrefs

Cf. A008280, A000111, A108040.

Programs

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

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 *)

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

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