A108040 -id:A108040 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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.

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.

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.

Cf. A008280, A000111, A108040.

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

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

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 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

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

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

A008280 Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, 61, 61, 56, 46, 32, 16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936

Offset: 0

N. J. A. Sloane

The earliest known reference for this triangle is Seidel (1877). - Don Knuth, Jul 13 2007

Sum of row n = A000111(n+1). - Reinhard Zumkeller, Nov 01 2013

			This version of the triangle begins:
  [0] [   1]
  [1] [   0,    1]
  [2] [   1,    1,    0]
  [3] [   0,    1,    2,    2]
  [4] [   5,    5,    4,    2,    0]
  [5] [   0,    5,   10,   14,   16,   16]
  [6] [  61,   61,   56,   46,   32,   16,    0]
  [7] [   0,   61,  122,  178,  224,  256,  272,  272]
  [8] [1385, 1385, 1324, 1202, 1024,  800,  544,  272,    0]
  [9] [   0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]
See A008281 and A108040 for other versions.

M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 110.
A. J. Kempner, On the shape of polynomial curves, Tohoku Math. J., 37 (1933), 347-362.
A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 53.
R. P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 141, Cambridge University Press (2012), p. 128, 174, 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
M. D. Atkinson, Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
Dominique Foata, Guo-Niu Han, and Volker Strehl, The Entringer-Poupard matrix sequence. Linear Algebra Appl. 512, 71-96 (2017). Example 4.3.
Ira M. Gessel, Counting up-up-or-down-down permutations, arXiv:2411.16113 [math.CO], 2024.
Boris Gourévitch, L'univers de Pi
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Christiane Poupard, De nouvelles significations énumératives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
Sanjay Ramassamy, Modular periodicity of the Euler numbers and a sequence by Arnold, arXiv:1712.08666 [math.CO], 2017.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187; see Beilage 5, pp. 183-184.
Ross Street, Trees, permutations and the tangent function, arXiv:math/0303267 [math.HO], 2003.
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A008281, A108040, A058257.

Cf. A000657 (central terms); A227862.

a008280 n k = a008280_tabl !! n !! k
a008280_row n = a008280_tabl !! n
a008280_tabl = ox True a008281_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
-- Reinhard Zumkeller, Nov 01 2013

max = 9; t[0, 0] = 1; t[n_, m_] /; n < m || m < 0 = 0; t[n_, m_] := t[n, m] = Sum[t[n-1, n-k], {k, m}]; tri = Table[t[n, m], {n, 0, max}, {m, 0, n}]; Flatten[ {Reverse[#[[1]]], #[[2]]} & /@ Partition[tri, 2]] (* Jean-François Alcover, Oct 24 2011 *)
T[0,0] := 1; T[n_?OddQ,k_]/;0<=k<=n := T[n,k]=T[n,k-1]+T[n-1,k-1]; T[n_?EvenQ,k_]/;0<= k<=n := T[n,k]=T[n,k+1]+T[n-1,k]; T[n_,k_] := 0; Flatten@Table[T[n,k], {n,0,9}, {k,0,n}] (* Oliver Seipel, Nov 24 2024 *)

T(n, m):=abs(sum(binomial(m, k)*euler(n-m+k), k, 0, m)); /* Vladimir Kruchinin, Apr 06 2015 */

# Python 3.2 or higher required.
from itertools import accumulate
A008280_list = blist = [1]
for n in range(10):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    A008280_list.extend(blist)
print(A008280_list) # Chai Wah Wu, Sep 20 2014

# Uses function seidel from A008281.
def A008280row(n): return seidel(n) if n % 2 else seidel(n)[::-1]
for n in range(8): print(A008280row(n)) # Peter Luschny, Jun 01 2022

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A008280_triangle(n) :
    A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        print([A[z] for z in (-i//2..i//2)])
A008280_triangle(10) # Peter Luschny, Jun 02 2012

T(n,m) = abs( Sum_{k=0..n} C(m,k)*Euler(n-m+k) ). - Vladimir Kruchinin, Apr 06 2015

E.g.f.: (cos(x) + sin(x))/cos(x+y). - Ira M. Gessel, Nov 18 2024

A005437 Column of Kempner tableau.

Original entry on oeis.org

1, 1, 1, 2, 4, 14, 46, 224, 1024, 6320, 36976, 275792, 1965664, 17180144, 144361456, 1446351104, 13997185024, 158116017920, 1731678144256, 21771730437632, 266182076161024, 3686171162253824, 49763143319190016, 752594181757712384, 11118629668610842624

Offset: 0

Simon Plouffe

nonn

From Peter Luschny, Jul 09 2012: (Start)
Also the central column of the Seidel-Entringer triangles A008281 and A008282.
a(n) takes alternatingly the values of the central column of the Seidel-Entriger triangles A008281 (1,1,4,46,...) and A008282 (1,2,14,224,..).
In Gelineau, Shin, and Zeng (section 6.1) twelve interpretations of the numbers can be found. (End)
This sequence is the central sequence of numbers in the following table:
A_0               1
B_1               1   0
A_2           0   1   1
B_3           2   2   1   0
A_4       0   2   4   5   5
B_5      16  16  14  10   5  0
A_6   0  16  32  46  56  61 61
B_7 272 272 256 224 178 122 61 0
where row A_k is obtained from row B_(k-1) by the sequence 0, b_1, b_1+b_2, ..., b_1+b_2+....+b_k and row B_k is obtained from the row A_(k-1) by the sequence a_1+a_2+....+a_k, ..., a_(k-1)+a_k, a_k, 0. - Sean A. Irvine, Jun 25 2016
Named after the English-American mathematician Aubrey John Kempner (1880-1973). - Amiram Eldar, Jun 23 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..485
Yoann Gelineau, Heesung Shin and Jiang Zeng, Bijections for Entringer families, hal-00472187, 2010.
Yoann Gelineau, Heesung Shin and Jiang Zeng, Bijections for Entringer families, arXiv:1004.2179 [math.CO], 2010.
Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Séminaire de théorie des nombres, 1980/1981, Exp. No. 11, p. 41, Univ. Bordeaux I, Talence, 1982.

Cf. A008281, A008282, A010094, A108040.

Main diagonal of A064192.

A005437 := proc(n) local S; S := proc(n, k) option remember; if k=0 then `if`(n=0, 1, 0) else S(n, k-1)+S(n-1, n-k) fi end: S(n, iquo(n+1, 2)) end; seq(A005437(i), i=0..24); # Peter Luschny, Jul 09 2012

a[n_] := Module[{S}, S[m_, k_] := S[m, k] = If[k == 0, If[m == 0, 1, 0], S[m, k-1] + S[m-1, m-k]]; S[n, Quotient[n+1, 2]]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 12 2018, after Peter Luschny *)

More terms from Sean A. Irvine, Jun 25 2016

Offset set to 0 by Peter Luschny, Oct 15 2018

A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.

Original entry on oeis.org

1, 0, 1, -1, -1, 0, 0, -1, -2, -2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, -61, -61, -56, -46, -32, -16, 0, 0, -61, -122, -178, -224, -256, -272, -272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936

Offset: 0

Paul Curtz, Mar 08 2014

			The triangle T(n,k) begins:
                      1
                    0   1
                 -1  -1   0
                0  -1  -2  -2
              5   5   4   2   0
             ...
The array read as a table, A(n,k) = T(n+k, k), starts:
     1,    1,    0,   -2,    0,   16,    0, -272,    0, ...
     0,   -1,   -2,    2,   16,  -16, -272,  272, ...
    -1,   -1,    4,   14,  -32, -256,  544, ...
     0,    5,   10,  -46, -224,  800, ...
     5,    5,  -56, -178, 1024, ...
     0,  -61, -122, 1202, ...
   -61,  -61, 1324, ...
     0, 1385, ...
  1385, ...
  ...
For the above table, we have A(n,k) = (-1)^(n+k)*A236935(n,k) for n, k >= 0. It has joint e.g.f. 2*exp(-x)/(1 + exp(-2*(x+y))). - _Petros Hadjicostas_, Feb 21 2021

L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).

Unsigned version is A008280.

Cf. A008281, A099023, A108040, A122045, A155585, A236935.

t[0, 0] = 1; t[n_, m_] /; nJean-François Alcover, Dec 30 2014 *)

T(n,m):=sum(binomial(m,k)*euler(n-m+k),k,0,m); /* Vladimir Kruchinin, Apr 06 2015 */

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
T(n, k) = (-1)^n*sum(i=0, k, (-1)^i*binomial(k, i)*a(n-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
T(n, k) = sum(i=0, k, binomial(k, i)*a(n-k+i)) /* Petros Hadjicostas, Feb 21 2021 */

a(n) = A057077(n)*A008280(n) by rows.
a(n) is the increasing antidiagonals of the difference table of A155585(n).
Central column of triangle: A099023(n).
Right main diagonal of triangle: A155585(n) (see A009006(n)).
Left main diagonal of triangle: A122045(n).
T(n,m) = Sum_{k=0..n} binomial(m,k)*Euler(n-m+k) for 0 <= m <= n. - Vladimir Kruchinin, Apr 06 2015 [The summation only needs to go from k=0 to k=m because of binomial(m,k).]
T(n,k) = (-1)^n*A236935(n-k,k) for 0 <= k <= n, where the latter is read as a square array. - Petros Hadjicostas, Feb 21 2021

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

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A008280 Boustrophedon version of triangle of Euler-Bernoulli or Entringer numbers read by rows.

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A005437 Column of Kempner tableau.

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A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.

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