A058257 -id:A058257 - cp's OEIS frontend

A058258 The 2-Up sequence: formed from final entries in rows of A058257.

1, 1, 1, 1, 3, 6, 26, 71, 413, 1456, 10576, 45541, 397023, 2020656, 20551376, 120686411, 1402815833, 9336345856, 122087570176, 908138776681, 13194844482843, 108480272749056, 1733786041150976, 15611712012050351, 272197308765744053, 2664103110372192256

Offset: 0

N. J. A. Sloane, Dec 06 2000

Alois P. Heinz, Table of n, a(n) for n = 0..200
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.

Cf. A005981, A076417.

Column k=2 of A229892.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=2,
       b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 02 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 2, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ;a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
CoefficientList[Series[1 + ((Sin[x]-Cos[x]+1) * (Cosh[x]-1) + (Sin[x]+Cos[x]+1) * Sinh[x]) / ((1+Cosh[x]*Cos[x])),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 06 2014 *)

E.g.f. (J. M. Luck, 2013): 1 + ((sin(x) - cos(x) + 1) * (cosh(x)-1) + (sin(x) + cos(x) + 1) * sinh(x)) / ((1 + cosh(x)*cos(x))). - Vaclav Kotesovec, Sep 06 2014

a(n) ~ c * n! / r^n, where r = A076417 = 1.8751040687119611664453... is the root of the equation cosh(r)*cos(r) = -1, and c = 4*cot(r/2)/r = 1.56598351207925... if n is even, c = 4*cot(r/2)^2/r = 1.14958147083780... if n is odd. - Vaclav Kotesovec, Sep 06 2014

More terms from Larry Reeves (larryr(AT)acm.org), Dec 12 2000

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

A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

Original entry on oeis.org

1, 6, 1456, 2020656, 9336345856, 108480272749056, 2664103110372192256, 122840808510269863827456, 9758611490955498257378246656, 1251231616578606273788469919481856, 245996119743058288132230759497577005056, 71155698830255977656506481145458378597728256

Offset: 0

Peter Bala, Jul 13 2007

Bisection of A005981.

			a(1) = 6. The six 2 up, 2 down permutations on 5 letters are (12543), (13542), (14532), (23541), (24532) and (34521).

Alois P. Heinz, Table of n, a(n) for n = 0..50
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Cf. A000111, A005981, A058257, A131454, A131455.

g:= (tan(x)+exp(2*x)*(tan(x)+1)-1)/(exp(2*x)+2*exp(x)*sec(x)+1): series(%,x,46):
seq(n!*coeff(%,x,n), n=1..45,4); # Peter Luschny, Feb 07 2017

Table[(CoefficientList[Series[((-1 + E^(2*x))*Cos[x] + (1 + E^(2*x))*Sin[x]) / (2*E^x + (1 + E^(2*x))* Cos[x]), {x, 0, 80}], x] * Range[0, 77]!)[[n]], {n, 2, 78, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+1))/(4n+1)! = (sin(x)*(exp(2x)+1)+cos(x)*(exp(2x)-1))/(2*exp(x)+cos(x)*(exp(2x)+1)).

A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

Original entry on oeis.org

1, 71, 45541, 120686411, 908138776681, 15611712012050351, 531909061958526321421, 32491881630252866646683891, 3302814916156503291298772711761, 527393971346575736206847604137659031, 126355819963625435928020023737689391659701

Offset: 0

Peter Bala, Jul 13 2007

Bisection of A005981. The entries listed above suggest various congruences for a(n): a(n) = 1 (mod 10), a(n) = 1 + 70*n (mod 100), a(n) = 1 + 70*n + 200*n(n-1) (mod 1000). Are these congruences true for all n? For an arbitrary integer m, the sequence a(n) taken modulo m may eventually become periodic. Compare with A081727.

			(1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Cf. A000111, A005981, A058257, A131453, A131455.

g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%,x,44):
seq(n!*coeff(%,x,n),n=3..45,4); # Peter Luschny, Feb 07 2017

Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+3))/(4n+3)! = (exp(2x)-2*sin(x)*exp(x)-1)/(2*exp(x)+cos(x)*(exp(2x)+1)). It appears that a(n) = (4n+3)!*coefficient of x^(4n+3) in the Taylor expansion of -4/(2*exp(x)+cos(x)*(exp(2x)+1)).

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A058258 The 2-Up sequence: formed from final entries in rows of A058257.

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

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A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

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A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

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