A062162 -id:A062162 - cp's OEIS frontend

A109449 Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1, 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1, 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1

Offset: 0

Philippe Deléham, Aug 27 2005

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.
The 'signed version' of the triangle is the exponential Riordan array [sech(x) + tanh(x), x]. - Peter Luschny, Jan 24 2009
Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 15 2013

			Triangle starts:
      1;
      1,     1;
      1,     2,     1;
      2,     3,     3,     1;
      5,     8,     6,     4,     1;
     16,    25,    20,    10,     5,    1;
     61,    96,    75,    40,    15,    6,    1;
    272,   427,   336,   175,    70,   21,    7,   1;
   1385,  2176,  1708,   896,   350,  112,   28,   8,  1;
   7936, 12465,  9792,  5124,  2016,  630,  168,  36,  9,  1;
  50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, The Swiss-Knife polynomials.
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).
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
See also : A000182, A000964, A009739, A062161, A062272.
Cf. A153641, A162660.
Cf. A000667 (row sums), A247453.

a109449 n k = a109449_row n !! k
a109449_row n = zipWith (*)
                (a007318_row n) (reverse $ take (n + 1) a000111_list)
a109449_tabl = map a109449_row [0..]
-- Reinhard Zumkeller, Nov 02 2013

f:= func< n,x | Evaluate(BernoulliPolynomial(n+1), x) >;
A109449:= func< n,k | k eq n select 1 else 2^(2*n-2*k+1)*Binomial(n,k)*Abs(f(n-k,3/4) - f(n-k,1/4) + f(n-k,1) - f(n-k,1/2))/(n-k+1) >;
[A109449(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jul 10 2025

From Peter Luschny, Jul 10 2009, edited Jun 06 2022: (Start)
A109449 := (n,k) -> binomial(n, k)*A000111(n-k):
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
B109449 := (n,k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k, 1)) -`if`(n-k=0, 1, 0): seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
sigma := n -> ifelse(n=0, 1, [1,1,0,-1,-1,-1,0,1][n mod 8 + 1]/2^iquo(n-1,2)-1):
L109449 := proc(n) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
X109449 := n -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9);
(End)

lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015, after Jean-François Alcover at A000111 *)
T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 27 2019 *)

A109449(n,k)=binomial(n,k)*if(n>k,2*abs(polylog(k-n,I)),1) \\ M. F. Hasler, Oct 05 2017

R = PolynomialRing(ZZ, 'x')
@CachedFunction
def skp(n, x) :
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A109449_row(n):
    x = R.gen()
    return [abs(c) for c in list(skp(n,x)-skp(n,x-1)+x^n)]
for n in (0..10) : print(A109449_row(n)) # Peter Luschny, Jul 22 2012

Sum_{k>=0} T(n, k) = A000667(n).
Sum_{k>=0} T(2n, 2k) = A000795(n).
Sum_{k>=0} T(2n, 2k+1) = A009747(n).
Sum_{k>=0} T(2n+1, 2k) = A003719(n).
Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
Sum_{k>=0} T(n, 2k) = A062272(n).
Sum_{k>=0} T(n, 2k+1) = A062161(n).
Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - Johannes W. Meijer, Apr 20 2011
E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic, May 20 2007
T(n,k) = 2^(n-k)*C(n,k)*|E(n-k,1/2) + E(n-k,1)| - [n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - Peter Luschny, Jan 24 2009
From Reikku Kulon, Feb 26 2009: (Start)
A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
A109449(n, n - 1) = n.
A109449(n, n) = 1.
For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
From Peter Luschny, Jul 10 2009: (Start)
Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)*F(k)*(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.
Another way to express the polynomials p_n(x) is
p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k)*Euler(k)((x+1)^(n-k) + x^(n-k)). (End)
From Peter Bala, Jan 26 2011: (Start)
An explicit formula for the n-th row polynomial is
x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.
Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)
Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then
T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - Peter Luschny, Jul 22 2012
T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - Reinhard Zumkeller, Nov 02 2013
T(n,k) = abs(A247453(n,k)). - Reinhard Zumkeller, Sep 17 2014

Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by Peter Luschny, Jul 10 2009

A062272 Boustrophedon transform of (n+1) mod 2.

Original entry on oeis.org

1, 1, 2, 5, 12, 41, 152, 685, 3472, 19921, 126752, 887765, 6781632, 56126201, 500231552, 4776869245, 48656756992, 526589630881, 6034272215552, 72989204937125, 929327412759552, 12424192360405961, 174008703107274752

Offset: 0

Frank Ellermann, Jun 16 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

A000734 (binomial transform), a(2n+1)= A003719(n), a(2n)= A000795(n),
Cf. A062161 (n mod 2).
Row sums of A162170 minus A000035. - Mats Granvik, Jun 27 2009
Cf. A059841.

a062272 n = sum $ zipWith (*) (a109449_row n) $ cycle [1,0]
-- Reinhard Zumkeller, Nov 03 2013

s[n_] = Mod[n+1, 2]; t[n_, 0] := s[n]; 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 *)

from itertools import accumulate, islice
def A062272_gen(): # generator of terms
    blist, m = tuple(), 0
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062272_list = list(islice(A062272_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062272_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == 1 else 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        R.append(A[e*i//2])
    return R
A062272_list(10) # Peter Luschny, Jun 02 2012

E.g.f.: (sec(x)+tan(x))cosh(x); a(n)=(A000667(n)+A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k)*A000111(n-2k). - Philippe Deléham, Aug 28 2005
a(n) = sum(A109449(n,k) * (1 - n mod 2): k=0..n). - Reinhard Zumkeller, Nov 03 2013

A062161 Boustrophedon transform of n mod 2.

Original entry on oeis.org

0, 1, 2, 4, 12, 36, 142, 624, 3192, 18256, 116282, 814144, 6219972, 51475776, 458790022, 4381112064, 44625674352, 482962852096, 5534347077362, 66942218896384, 852334810990332, 11394877025289216, 159592488559874302, 2336793875186479104, 35703580441464231912

Offset: 0

Frank Ellermann, Jun 10 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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).
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Cf. A000035, A062272.

a062161 n = sum $ zipWith (*) (a109449_row n) $ cycle [0,1]
-- Reinhard Zumkeller, Nov 03 2013

With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])Sinh[x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Feb 16 2013 *)

from itertools import accumulate, islice
def A062161_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062161_list = list(islice(A062161_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062161_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == -1 else 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)]
        R.append(A[e*i//2])
    return R
A062161_list(10) # Peter Luschny, Jun 02 2012

a(2n) = A009747(n), a(2n+1) = A002084(n).
E.g.f.: (sec(x)+tan(x))*sinh(x); a(n)=(A000667(n)-A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k+1)*A000111(n-2k-1). - Philippe Deléham, Aug 28 2005
a(n) = Sum_{k=0..n} A109449(n,k) * (k mod 2). - Reinhard Zumkeller, Nov 03 2013 [corrected by Jason Yuen, Jan 07 2025]

A247453 T(n,k) = binomial(n,k)A000111(n-k)(-1)^(n-k), 0 <= k <= n.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 5, -8, 6, -4, 1, -16, 25, -20, 10, -5, 1, 61, -96, 75, -40, 15, -6, 1, -272, 427, -336, 175, -70, 21, -7, 1, 1385, -2176, 1708, -896, 350, -112, 28, -8, 1, -7936, 12465, -9792, 5124, -2016, 630, -168, 36, -9, 1, 50521

Offset: 0

Reinhard Zumkeller, Sep 17 2014

Matrix inverse of A109449, the unsigned version of this sequence. More precisely, consider both of these triangles as the nonzero lower left of an infinite square array / matrix, filled with zeros above/right of the diagonal. Then these are mutually inverse of each other; in matrix notation: A247453 . A109449 = A109449 . A247453 = Identity matrix. In more conventional notation, for any m,n >= 0, Sum_{k=0..n} A247453(n,k)*A109449(k,m) = Sum_{k=0..n} A109449(n,k)*A247453(k,m) = delta(m,n), the Kronecker delta (= 1 if m = n, 0 else). - M. F. Hasler, Oct 06 2017

			.   0:      1
.   1:     -1      1
.   2:      1     -2      1
.   3:     -2      3     -3      1
.   4:      5     -8      6     -4      1
.   5:    -16     25    -20     10     -5     1
.   6:     61    -96     75    -40     15    -6     1
.   7:   -272    427   -336    175    -70    21    -7    1
.   8:   1385  -2176   1708   -896    350  -112    28   -8   1
.   9:  -7936  12465  -9792   5124  -2016   630  -168   36  -9   1
.  10:  50521 -79360  62325 -32640  12810 -4032  1050 -240  45 -10  1  .

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
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).
OEIS Wiki, Boustrophedon transform.
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A007318, A062162, A109449.

a247453 n k = a247453_tabl !! n !! k
a247453_row n = a247453_tabl !! n
a247453_tabl = zipWith (zipWith (*)) a109449_tabl a097807_tabl

a111[n_] := n! SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n}];
T[n_, k_] := (-1)^(n-k) Binomial[n, k] a111[n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

A247453(n,k)=(-1)^(n-k)*binomial(n,k)*if(n>k, 2*abs(polylog(k-n, I)), 1) \\ M. F. Hasler, Oct 06 2017

T(n,k) = (-1)^(n-k) * A007318(n,k) * A000111(n-k), k = 0..n;
T(n,k) = (-1)^(n-k) * A109449(n,k); A109449(n,k) = abs(T(n,k));
abs(sum of row n) = A062162(n);
Sum_{k=0..n} T(n,k)*A000111(k) = 0^n.

Edited by M. F. Hasler, Oct 06 2017

A101473 Boustrophedon transform of the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 3, 9, 31, 111, 453, 2059, 10571, 60651, 386253, 2704659, 20661411, 170990691, 1523975053, 14552848059, 148234015051, 1604267622731, 18383552327853, 222363321668259, 2831217743661491, 37850593064646771, 530121590756400653

Offset: 0

Paul Barry, Jan 21 2005

Binomial transform of A101474.

Cf. A000752, A001045, A062162, A101474.

from itertools import accumulate, islice
def A101473_gen(): # generator of terms
    blist, a, b = tuple(), 0, 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
        a, b = b, 2*a+b
A101473_list = list(islice(A101473_gen(),30)) # Chai Wah Wu, Jun 11 2022

E.g.f.: (sec(x) + tan(x))*(exp(2*x) - exp(-x))/3.
a(n) = (A000752(n) - A062162(n))/3.
a(n) ~ n! * 2^(n+2) * (exp(Pi) - exp(-Pi/2)) / (3 * Pi^(n+1)). - Vaclav Kotesovec, Jun 12 2015

A102590 Inverse Boustrophedon transform of 2^n.

Original entry on oeis.org

1, 1, 1, 0, -3, -14, -39, -130, -263, -1214, -179, -21810, 98277, -1021214, 8446881, -82814290, 836117617, -9075846014, 103898533141, -1257148371570, 16004750729757, -213975589371614, 2996827456610601, -43880489398997650, 670443584312526697, -10670445866332254014

Offset: 0

Paul Barry, Jan 22 2005

Binomial transform of (-1)^n*A062162.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000079, A062162.

a:= n-> n!*coeff(series(exp(2*x)/(sec(x)+tan(x)), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 29 2013

CoefficientList[Series[Cos[x]*E^(2*x)/(1+Sin[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)

from itertools import islice, accumulate
from operator import sub
def A102590_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=m)))[-1]
        m *= 2
A102590_list = list(islice(A102590_gen(),20)) # Chai Wah Wu, Jun 10 2022

E.g.f.: exp(2x)/(sec(x)+tan(x)) = cos(x)exp(2x)/(1+sin(x)).
a(n) ~ (-1)^n * n^(n+1/2)*2^(n+5/2)/(Pi^(n+1/2)*exp(n+Pi)). - Vaclav Kotesovec, Sep 29 2013
G.f.: E(0)*x/(x-1)/(1-2*x) + 1/(1-2*x), where E(k) = 1 - x^2*(k+1)*(k+2)/( x^2*(k+1)*(k+2) - 2*(x*(k-1)+1)*(x*k+1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014

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

A363394 Triangle read by rows. T(n, k) = A081658(n, k) + A363393(n, k) for k > 0 and T(n, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, -1, 1, 3, -3, -2, 1, 4, -6, -8, 5, 1, 5, -10, -20, 25, 16, 1, 6, -15, -40, 75, 96, -61, 1, 7, -21, -70, 175, 336, -427, -272, 1, 8, -28, -112, 350, 896, -1708, -2176, 1385, 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936

Offset: 0

Peter Luschny, Jun 06 2023

			The triangle T(n, k) begins:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  -1;
  [3] 1, 3,  -3,   -2;
  [4] 1, 4,  -6,   -8,   5;
  [5] 1, 5, -10,  -20,  25,   16;
  [6] 1, 6, -15,  -40,  75,   96,   -61;
  [7] 1, 7, -21,  -70, 175,  336,  -427,  -272;
  [8] 1, 8, -28, -112, 350,  896, -1708, -2176,  1385;
  [9] 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936;

Richard P. Stanley, A survey of alternating permutations, arXiv:0912.4240 [math.CO], 2009.
Index entries for sequences related to Euler numbers.

Variants (row reversed): A109449, A247453.

Cf. A081658 (signed secant part), A363393 (signed tangent part), A000111 (main diagonal), A122045, A155585 (aerated main diagonal), A000667, A062162 (row sums of signless variant).

# Variant, computes abs(T(n, k)):
P := n -> n!*coeff(series((sec(y) + tan(y))/exp(x*y), y, 24), y, n):
seq(print(seq((-1)^(n - k)*coeff(P(n), x, n - k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k == n:
        p = k % 2
        return p - sum(T(n, j) for j in range(p, n - 1, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 1)])

|T(n, k)| = (-1)^(n - k) * n! * [x^(n - k)][y^n] (sec(y) + tan(y)) / exp(x*y).
T(n, k) = [x^(n - k)] -2^(k-(0^k))*(Euler(k, 0) + Euler(k, 1/2)) / (x-1)^(k + 1).
For a recursion see the Python program.
T(n, k) = [x^n] ((-1) + Sum_{j=0..n} binomial(n, j)*(Euler(j, 1) + Euler(j, 1/2))*(2*x)^j). - Peter Luschny, Nov 17 2024

cp's OEIS Frontend

A109449 Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

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A062272 Boustrophedon transform of (n+1) mod 2.

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A062161 Boustrophedon transform of n mod 2.

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A247453 T(n,k) = binomial(n,k)*A000111(n-k)*(-1)^(n-k), 0 <= k <= n.

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A101473 Boustrophedon transform of the Jacobsthal numbers.

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A102590 Inverse Boustrophedon transform of 2^n.

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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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A247453 T(n,k) = binomial(n,k)A000111(n-k)(-1)^(n-k), 0 <= k <= n.