A000657 -id:A000657 - cp's OEIS frontend

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

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

A005799 Generalized Euler numbers of type 2^n.

Original entry on oeis.org

1, 1, 2, 10, 104, 1816, 47312, 1714000, 82285184, 5052370816, 386051862272, 35917232669440, 3996998043812864, 524203898507631616, 80011968856686405632, 14061403972845412526080, 2818858067801804443910144

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also, a(n) equals the number of alternating permutations (p(1),...,p(2n)) of the multiset {1,1,2,2,...,n,n} satisfying p(1) <= p(2) > p(3) <= p(4) > p(5) <= ... <= p(2n). Hence, A275801(n) <= a(n) <= A275829(n). - Max Alekseyev, Aug 10 2016
This is the BinomialMean transform of A000364 (see A075271 for definition of transform). - John W. Layman, Dec 04 2002
This sequence appears to be middle column in Poupard's triangle A008301.

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

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Proof of several conjectures relating permanents to Combinatorial sequences, arXiv:2109.11506v3 [math.CO], 2021-2023.
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Permanent identities, combinatorial sequences, and permutation statistics, Advances in Applied Mathematics, Volume 163, Part A, 102789 (2025).
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
H. Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Right edge of triangle A210108.

Cf. A000364, A008301, A275801, A275829, A154283, A000657.

T := proc(n, k) option remember;
if n < 0 or k < 0 then 0
elif n = 0 then euler(k, 1)
else T(n-1, k+1) - T(n-1, k) fi end:
a := n -> (-2)^n*T(n, n); seq(a(n), n=0..16); # Peter Luschny, Aug 23 2017

a[n_] := Sum[Binomial[n, i]Abs[EulerE[2i]], {i, 0, n}]/2^n

a(n) = (1/2^n) * Sum_{i=0..n} binomial(n, i) * A000364(i).
From Sergei N. Gladkovskii, Dec 27 2012, Oct 11 2013, Oct 27 2013, Jan 08 2014: (Start) Continued fractions:
G.f.: A(x) = 1/G(0) where G(k) = 1 - x*(k+1)*(2*k+1)/(1 - x*(k+1)*(2*k+1)/G(k+1)).
G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2*(2*k+1)^2/(x^2*(k+1)^2*(2*k+1)^2 - (4*x*k^2 + 2*x*k + x - 1)*( 4*x*k^2 + 10*x*k + 7*x - 1)/Q(k+1)).
G.f.: R(0), where R(k) = 1 - x*(2*k+1)*(k+1)/(x*(2*k+1)*(k+1) - 1/(1 - x*(2*k+1)*(k+1)/(x*(2*k+1)*(k+1) - 1/R(k+1)))).
G.f.: 2/(x*Q(0)), where Q(k) = 2/x - 1 - (2*k+1)^2/(1 - (2*k+2)^2/Q(k+1)). (End)
a(n) ~ 2^(3*n+3) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, May 30 2015
a(n) = 2^n * Sum_{k=0..n} (-1)^k*binomial(n, k)*euler(n+k, 1). - Peter Luschny, Aug 23 2017
From Peter Bala, Dec 21 2019: (Start)
O.g.f. as a continued fraction: 1/(1 - x/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 15*x/(1 - 15*x/(1 - ... - n*(2*n-1)*x/(1 - n*(2*n-1)*x/(1 - ...))))))))) - apply Bala, Proposition 3, with a = 0, b = 1 and replace x with x/2.
Conjectures:
E.g.f. as a continued fraction: 2/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t + 2*t^2/2! + 10*t^3/3! + 104*t^4/4! + ....
Cf. A000657. [added April 18 2024: for a proof of this conjecture see Fu et al., Section 4.3.]
a(n) = (-2)^(n+1)*Sum_{k = 0..floor((n-1)/2)} binomial(n,2*k+1)*(2^(2*n-2*k) - 1)*Bernoulli(2*n-2*k)/(2*n-2*k) for n >= 1. (End)

Edited by Dean Hickerson, Dec 10 2002

A002832 Median Euler numbers.

Original entry on oeis.org

1, 3, 24, 402, 11616, 514608, 32394624, 2748340752, 302234850816, 41811782731008, 7106160248346624, 1455425220196234752, 353536812021243273216, 100492698847094242603008, 33045185784774350171111424

Offset: 1

N. J. A. Sloane, Dec 11 1996

nonn

There are two kinds of Euler median numbers, the 'right' median numbers (this sequence), and the 'left' median numbers (A000657).

Apparently all terms (except the initial 1) have 3-valuation 1. - F. Chapoton, Aug 02 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26. (In French, with a summary in English on p. 1).
R. C. Read, Letter to N. J. A. Sloane, 1992

Cf. A000657.
See related polynomials in A098277.
A diagonal of A323833.

rr := array(1..40,1..40):rr[1,1] := 0:for i from 1 to 39 do rr[i+1,1] := (subs(x=0,diff((exp(x)-1)/cosh(x),x$i))):od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-rr[i-1,j-1]:od:od: seq(rr[2*i-1,i-1],i=2..20); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 16 2001, corrected by R. J. Mathar, Dec 22 2010
# alternative
A002832 := proc(n)
    abs(A323833(n-1,n)) ;
end proc:
seq(A002832(n),n=1..40) ; # R. J. Mathar, Jun 11 2025

max = 20; rr[1, 1] = 0; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[(Exp[x] - 1)/Cosh[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - rr[i - 1, j - 1]]]; Table[(-1)^i*rr[2*i - 1, i - 1], {i, 2, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)

G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*3x/(1-1*5x/(1-2*7x/(1-2*9x/(1-3*11x/...))))).
G.f.: -1/G(0) where G(k)= x*(8*k^2+8*k+3) - 1 - (4*k+5)*(4*k+3)*(k+1)^2*x^2/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 08 2012
a(n) ~ 2^(4*n+3/2) * n^(2*n-1/2) / (exp(2*n) * Pi^(2*n-1/2)). - Vaclav Kotesovec, Apr 23 2015

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001

Terms corrected by R. J. Mathar, Dec 22 2010

A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).

Original entry on oeis.org

1, -1, 4, -46, 1024, -36976, 1965664, -144361456, 13997185024, -1731678144256, 266182076161024, -49763143319190016, 11118629668610842624, -2925890822304510631936, 895658946905031792553984

Offset: 0

Ralf Stephan, Sep 23 2004

sign

T(2n,n), where T is A008280 (signed).

Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A000657, A008280, A029582, A009744.

A099023List[n_] := Module[{e, dim, m, k}, dim = 2 n; e[0, 0] = 1; For[m = 1, m <= dim - 1, m++, If[EvenQ[m], e[m, 0] = 1; For[k = m - 1, k >= -1, k--, e[k, m - k] = e[k + 1, m - k - 1] - e[k, m - k - 1]], e[0, m] = 1; For[k = 1, k <= m + 1, k++, e[k, m - k] = e[k - 1, m - k + 1] + e[k - 1, m - k]]]]; Table[e[k, k], {k, 0, (dim + 1)/2 - 1}]];
A099023List[15] (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

# Variant of an algorithm of L. Seidel (1877).
def A099023_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0,0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m,0] = 1;
            for k in range(m-1,-1,-1) :
                E[k,m-k] = E[k+1,m-k-1] - E[k,m-k-1]
        else :
            E[0,m] = 1;
            for k in range(1,m+1,1) :
                E[k,m-k] = E[k-1,m-k+1] + E[k-1,m-k]
    return [E[k,k] for k in range((dim+1)//2)]
# Peter Luschny, Jul 14 2012

|a(n)| = A000657(n) - Sean A. Irvine, Dec 22 2010
G.f.: 1/G(0) where G(k) = 1 + x*(k+1)*(4*k+1)/(1 + x*(k+1)*(4*k+3)/G(k+1) ) ;  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1+x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 + x*(8*k^2+4*k+1))*(1 + x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014

A098277 Coefficients of polynomials D(n,x) related to median Euler numbers.

Original entry on oeis.org

1, 2, 2, 8, 20, 12, 48, 224, 344, 168, 384, 2880, 8096, 9872, 4272, 3840, 42240, 186816, 407936, 430688, 171168, 46080, 698880, 4451328, 15030528, 27944576, 26627648, 9915072, 645120, 12902400, 111605760, 535271424, 1519126272

Offset: 0

Ralf Stephan, Sep 07 2004

2^n(x+1) divides D(n,x).

			D(0,x) = 1,
D(1,x) = 2*x + 2,
D(2,x) = 8*x^2 + 20*x + 12,
D(3,x) = 48*x^3 + 224*x^2 + 344*x + 168,
D(4,x) = 384*x^4 + 2880*x^3 + 8096*x^2 + 9872*x + 4272.

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.

D(n, 1/2) = A002832(n+1), D(n, -1/2) = A000657(n).
D(n, 0)/2^n = A098278(n), D(n, 1)/2^n = A098279(n).
Leading coefficients are A000165. Constant terms are in A098431.

d[0, ] = 1; d[n, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2] - x(x+1)d[n-1, x];
Table[CoefficientList[d[n, x], x] // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)

D(n,x)=if(n<1,1,(x+1)*(x+2)*D(n-1,x+2)-x*(x+1)*D(n-1,x))

T(n,k)=local(A=sum(m=0,n,m!*(2*x)^m*prod(j=1,m,(j+y)/(1+j*(j+1)*x +x*O(x^n)))));polcoeff(polcoeff(A,n,x),n-k,y)
{for(n=0,8,for(k=0,n,print1(T(n,k),", "));print())} \\ Paul D. Hanna, Sep 05 2012

Recurrence: D(0, x)=1, D(n, x) = (x+1)(x+2)D(n-1, x+2) - x(x+1)D(n-1, x).
G.f.: Sum[n>=0, D(n, x)t^n] = 1/(1-2(x+1)t/(1-2(x+2)t/(1-4(x+3)t/(1-4(x+4)t/...)))).
G.f.: Sum_{n>=0} D(n,y)*x^n = Sum_{n>=0} n!*(2*x)^n*Product_{k=1..n} (k+y)/(1+k*(k+1)*x). - Paul D. Hanna, Sep 05 2012

A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

Original entry on oeis.org

1, 1, -1, -5, 5, 61, -61, -1385, 1385, 50521, -50521, -2702765, 2702765, 199360981, -199360981, -19391512145, 19391512145, 2404879675441, -2404879675441, -370371188237525, 370371188237525, 69348874393137901, -69348874393137901, -15514534163557086905

Offset: 0

Paul Curtz, Apr 17 2014

sign

A version of the Seidel triangle (1877) for the integer Euler numbers is
              1
           1    1
         2    2   1
       2   4    5   5
    16  16   14  10   5
  16  32  46   56  61  61
etc.
It is not in the OEIS. See A008282.
The first diagonal, Es(n) = 1, 1, 1, 5, 5, 61, 61, 1385, 1385, ..., comes from essentially A000364(n) repeated.
a(n) is Es(n) signed two by two.
Difference table of a(n):
     1,    1,   -1,   -5,     5,    61,   -61, -1385, ...
     0,   -2,   -4,   10,    56,  -122, -1324, ...
    -2,   -2,   14,   46,  -178, -1202, ...
     0,   16,   32, -224, -1024, ...
    16,   16, -256, -800, ...
     0, -272, -544, ...
  -272, -272, ...
     0, ...
etc.
Sum of the antidiagonals: 1, 1, -5, -11, 61, 211, -385, ... = A239322(n+1).
Main diagonal interleaved with the first upper diagonal: 1, 1, -2, -4, 14, 46, ... = signed A214267(n+1).
Inverse binomial transform (first column): A155585(n+1).
The Akiyama-Tanigawa transform applied to A046978(n+1)/A016116(n) gives
    1,   1,   1/2,   0, -1/4, -1/4, -1/8, 0, ...
    0,   1,   3/2,   1,    0, -3/4, -7/8, ...
   -1,  -1,   3/2,   4, 15/4,  3/4, ...
    0,  -5, -15/2,   1,   15, ...
    5,   5, -51/2, -56, ...
    0,  61, 183/2, ...
  -61, -61, ...
    0, ...
etc.
A122045(n) and A239005(n) are reciprocal sequences by their inverse binomial transform. In their respective difference table, two different signed versions of A214247(n) appear: 1) interleaved main diagonal and first under diagonal (1, -1, -1, 2, 4, -14, ...) and 2) interleaved main diagonal and first upper diagonal (1, 1, -1, -2, 4, 14, ...).

G. C. Greubel, Table of n, a(n) for n = 0..475

Cf. A000364, A000657, A008282, A016116, A046978, A119880, A122045, A153641, A155585, A214247, A214267, A239005, A239322.

EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
Euler:= func< n | 2^n*EulerPoly(n, 1/2) >; // A122045
[Euler(n) - Euler(n+1): n in [0..40]]; // G. C. Greubel, Jun 07 2023

A241209 := proc(n) local v, k, h, m; m := `if`(n mod 2 = 0, n, n+1);
h := k -> `if`(k mod 4 = 0, 0, (-1)^iquo(k,4));
(-1)^n*add(2^iquo(-k,2)*h(k+1)*add((-1)^v*binomial(k,v)*(v+1)^m, v=0..k)
,k=0..m) end: seq(A241209(n),n=0..24); # Peter Luschny, Apr 17 2014

skp[n_, x_]:= Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}];
a[n_]:= skp[n, x] - skp[n+1, x]/. x->0; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 17 2014, after Peter Luschny *)
Table[EulerE[n] - EulerE[n+1], {n,0,30}] (* Vincenzo Librandi, Jan 24 2016 *)
-Differences/@Partition[EulerE[Range[0,30]],2,1]//Flatten (* Harvey P. Dale, Apr 16 2019 *)

[euler_number(n) - euler_number(n+1) for n in range(41)] # G. C. Greubel, Jun 07 2023

a(n) = A119880(n+1) - A119880(n).
a(n) is the second column of the fractional array.
a(n) = (-1)^n*second column of the array in A239005(n).
a(n) = skp(n, 0) - skp(n+1, 0), where skp(n, x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 17 2014
E.g.f.: exp(x)/cosh(x)^2. - Sergei N. Gladkovskii, Jan 23 2016
G.f. T(0)/x-1/x, where T(k) = 1 - x*(k+1)/(x*(k+1)-(1-x)/(1-x*(k+1)/(x*(k+1)+(1-x)/T(k+1)))). - Sergei N. Gladkovskii, Jan 23 2016

A323834 A Seidel matrix A(n,k) read by antidiagonals downwards.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, -2, -1, 1, 4, -5, -7, -8, -7, -3, 16, 11, 4, -4, -11, -14, 61, 77, 88, 92, 88, 77, 63, -272, -211, -134, -46, 46, 134, 211, 274, -1385, -1657, -1868, -2002, -2048, -2002, -1868, -1657, -1383, 7936, 6551, 4894, 3026, 1024, -1024, -3026, -4894, -6551, -7934, 50521, 58457, 65008, 69902, 72928, 73952, 72928, 69902, 65008, 58457, 50523

Offset: 0

N. J. A. Sloane, Feb 03 2019

The first row is a signed version of the Euler numbers A000111.

Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).

			Read as triangle T(n,k) = A(k, n-k) (n >= 0, k = 0..n), the first few antidiagonals of the square array A are:
     0,
     1,    1,
     1,    2,    3,
    -2,   -1,    1,   4,
    -5,   -7,   -8,  -7,  -3,
    16,   11,    4,  -4, -11, -14,
    61,   77,   88,  92,  88,  77,  63,
  -272, -211, -134, -46,  46, 134, 211, 274,
  ...
From _Petros Hadjicostas_, Mar 02 2021: (Start)
Square array A(n,k) (with rows n >= 0 and columns k >= 0) begins:
     0,   1,   1,    -2,    -5,    16,     61,     -272,    -1385, ...
     1,   2,  -1,    -7,    11,    77,   -211,    -1657,     6551, ...
     3,   1,  -8,     4,    88,  -134,  -1868,     4894,    65008, ...
     4,  -7,  -4,    92,   -46, -2002,   3026,    69902,  -179806, ...
    -3, -11,  88,    46, -2048,  1024,  72928,  -109904, -3784448, ...
   -14,  77, 134, -2002, -1024, 73952, -36976, -3894352,  5860016, ...
   ... (End)

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (matrix b_{n,k} on p. 19).

Cf. A000111, A000657 (next-to-main diagonal), A323833.

{b(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ Michael Somos's PARI program for A000111.
c(n) = if(n==0, 0, (-1)^floor((n-1)/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 02 2021

From Petros Hadjicostas, Mar 02 2021: (Start)
Formulas for the square array A(n,k) (n, k >= 0):
A(0,k) = (-1)^floor((k-1)/2)*A000111(k) for k > 0 with A(0,0) = 0.
A(n,k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i) for n, k >= 0.
A(n,n)/2 = A(n+1,n) = +/- A000657(n) for n > 0.
Bivariate e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = (-sech(x + y) + tanh(x + y) + 1)*exp(x).
Formulas for the triangular array T(n,k) = A(k,n-k) (n >= 0, 0 <= k <= n):
T(n,k) = T(n-1,k-1) + T(n,k-1) for 1 <= k <= n with T(n,0) = (-1)^floor((n-1)/2) * A000111(n) for n > 0 and T(0,0) = 0.
T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-k+i,0) for 0 <= k <= n. (End)

Typo corrected by and more terms from Petros Hadjicostas, Mar 02 2021

A135594 a(n) = (1/2^n) * Sum_{i=0..n} (-1)^(n-i) * binomial(n,i) * A000364(i).

Original entry on oeis.org

1, 0, 1, 6, 73, 1380, 37801, 1417626, 69802993, 4369750440, 339034806001, 31935510092046, 3590398569115513, 474937566660074700, 73024143791301120601, 12914495107705743175266, 2603190607000627341985633, 593297406341867021292734160

Offset: 0

Vladeta Jovovic, Feb 25 2008

Let k be a positive integer. It appears that reducing the sequence {a(n): n >= 1} modulo k produces a periodic sequence with period a divisor of phi(k) unless k is of the form 2^j, when the period equals k. For example, modulo 7 the sequence becomes [0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, ...], with an apparent period of 6 = phi(7), while modulo 8 the sequence becomes [0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, ...] with an apparent period of 8. - Peter Bala, May 07 2023

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 4.2.2.(b).

Cf. A000657, A005799.

A000364 := proc(n) option remember ; (2*n)!*coeftayl(sec(x),x=0,2*n) ; end: A135594 := proc(n) add((-1)^(n-i)*binomial(n,i)*A000364(i),i=0..n)/2^n ; end: seq(A135594(n),n=0..20) ; # R. J. Mathar, Mar 14 2008
f:=sec(z): fser:=series(f,z=0,63): for n from 0 to 60 do b[n]:=factorial(n)*coeff(fser,z,n) end do: a:= proc(n) options operator, arrow: add((-1)^(n-k)*binomial(n,k)*b[2*k],k=0..n)/2^n end proc: seq(a(n),n=0..16); # Emeric Deutsch, Mar 17 2008

Table[(-1)^n*Sum[Binomial[n, k]*EulerE[2*k], {k, 0, n}]/2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jun 08 2019 *)

G.f.: 1/Q(0), where Q(k)= 1 + x - x*(2*k+1)*(k+1)/(1 - x*(2*k+1)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ 2^(3*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019
Conjecture: e.g.f. as a continued fraction: 2*exp(-t)/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t^2/2! + 6*t^3/3! + 73*t^4/4! + .... Cf. A000657 and A005799. - Peter Bala, Dec 21 2019

More terms from R. J. Mathar and Emeric Deutsch, Mar 03 2008

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