A227862 -id:A227862 - cp's OEIS frontend

A000667 Boustrophedon transform of all-1's sequence.

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Offset: 0

N. J. A. Sloane and Simon Plouffe

Fill in a triangle, like Pascal's triangle, beginning each row with a 1 and filling in rows alternately right to left and left to right.

Row sums of triangle A109449. - Reinhard Zumkeller, Nov 04 2013

			...............1..............
............1..->..2..........
.........4..<-.3...<-..1......
......1..->.5..->..8...->..9..

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig 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. [USA access only through the HATHI TRUST Digital Library]
Ludwig 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. [Access through ZOBODAT]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Absolute value of pairwise sums of A009337.
Column k=1 of A292975.
Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.

a000667 n = if x == 1 then last xs else x
            where xs@(x:_) = a227862_row n
-- Reinhard Zumkeller, Nov 01 2013

With[{nn=30},CoefficientList[Series[Exp[x](Tan[x]+Sec[x]),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 28 2011 *)
t[, 0] = 1; t[n, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
a[n_] := t[n, n];
Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ Joerg Arndt, Jul 30 2016

from itertools import islice, accumulate
def A000667_gen(): # generator of terms
    blist = tuple()
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=1)))[-1]
A000667_list = list(islice(A000667_gen(),20)) # Chai Wah Wu, Jun 11 2022

# Algorithm of L. Seidel (1877)
def A000667_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1
        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)]
        R.append(A[e*i//2])
    return R
A000667_list(10)  # Peter Luschny, Jun 02 2012

E.g.f.: exp(x) * (tan(x) + sec(x)).
Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi; lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1 + 1/(n-1). - Gerald McGarvey, Aug 13 2004
a(n) = Sum_{k=0..n} binomial(n, k)*A000111(n-k). a(2*n) = A000795(n) + A009747(n), a(2*n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005
a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013
G.f.: E(0)*x/(1-x)/(1-2*x) + 1/(1-x), where E(k) = 1 - x^2*(k + 1)*(k + 2)/(x^2*(k + 1)*(k + 2) - 2*(x*(k + 2) - 1)*(x*(k + 3) - 1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

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

A261880 Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.

Original entry on oeis.org

1, -1, -2, 1, 2, 4, -2, -3, -5, -9, 5, 7, 10, 15, 24, -16, -21, -28, -38, -53, -77, 61, 77, 98, 126, 164, 217, 294, -272, -333, -410, -508, -634, -798, -1015, -1309, 1385, 1657, 1990, 2400, 2908, 3542, 4340, 5355, 6664

Offset: 0

Paul Curtz, Jul 10 2016

Difference array of (-1)^n*A000111(n):
1, -1, 1, -2, 5, ...
-2, 2, -3, 7,...
4, -5, 10, ...
-9, 15, ...
24, ... .
First column:(-1)^n*A000667(n).
Antidiagonal sums: b(n) =  1, -3, 7, -19, 61, -233, 1037, -5279, 30241, ..., i.e., row sums of the triangle.
Any triangle with entries T(n, m) built from some sequence in column m=0, and the recurrence T(n, m) = T(n, m-1) - T(n-1, m-1) for m >= 1, has the property that the new triangle t(n, m) = T(n+1, m+1) - T(n+1, m), 0 <= m <= n, equals -T(n, m). See the question in the example. - Wolfdieter Lang, Aug 08 2016

			The triangle T(n, m) begins:
n\m  0   1   2   3   4   5 ...
0:   1
1:  -1  -2
2:   1   2   4
3:  -2  -3  -5  -9
4:   5   7  10  15  24,
5: -16 -21 -28 -38 -53 -77
...
Triangle of differences of the row entries of the preceding triangle starting with row n=1:
n\m  0   1    2   3   4 ...
0:  -1
1:   1   2
2:  -1  -2   -4
3:   2   3    5   9
4:  -5  -7  -10 -15 -24
... .
This is the negative of the first triangle. Are there other sequences with the same property?

Cf. A000111, A000667, A062162, A227862, A239005.

Recurrence: T(n, 0) = (-1)^n*A000111(n), n >= 0. T(n, m) = T(n, m-1) - T(n-1, m-1), m >= 1. (from the fact that the differences of the rows, starting with n = 1 produce the negative of the triangle. See the example and a comment). - Wolfdieter Lang, Aug 08 2016

Edited by Wolfdieter Lang, Aug 08 2016

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A000667 Boustrophedon transform of all-1's sequence.

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

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A261880 Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.

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