A060097 -id:A060097 - cp's OEIS frontend

A198631 Numerators of the rational sequence with e.g.f. 1/(1+exp(-x)).

1, 1, 0, -1, 0, 1, 0, -17, 0, 31, 0, -691, 0, 5461, 0, -929569, 0, 3202291, 0, -221930581, 0, 4722116521, 0, -968383680827, 0, 14717667114151, 0, -2093660879252671, 0, 86125672563201181, 0, -129848163681107301953, 0, 868320396104950823611, 0

Offset: 0

Wolfdieter Lang, Oct 31 2011

Numerators of the row sums of the Euler triangle A060096/A060097.
The corresponding denominator sequence looks like 2*A006519(n+1) when n is odd.
Also numerator of the value at the origin of the n-th derivative of the standard logistic function. - Enrique Pérez Herrero, Feb 15 2016

			The rational sequence r(n) = a(n) / A006519(n+1) starts:
1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8, 0, 31/2, 0, -691/4, 0, 5461/2, 0, -929569/16, 0, 3202291/2, 0, -221930581/4, 0, 4722116521/2, 0, -968383680827/8, 0, 14717667114151/2, 0, -2093660879252671/4, ...

Robert Israel, Table of n, a(n) for n = 0..550
Eric Weisstein's World of Mathematics, Sigmoid Function.
Wikipedia, Logistic Function.

Cf. A000182, A060096, A060097, A006519, A002425, A089171, A090681.

seq(denom(euler(i,x))*euler(i,1),i=0..33); # Peter Luschny, Jun 16 2012

Join[{1},Table[Numerator[EulerE[n,1]/(2^n-1)], {n, 34}]] (* Peter Luschny, Jul 14 2013 *)

def A198631_list(n):
    x = var('x')
    s = (1/(1+exp(-x))).series(x,n+2)
    return [(factorial(i)*s.coefficient(x,i)).numerator() for i in (0..n)]
A198631_list(34) # Peter Luschny, Jul 12 2012

# Alternatively:
def A198631_list(len):
    e, f, R, C = 2, 1, [], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = -C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(numerator((e-1)*f*C[0]))
        f *= n; e <<= 1
    return R
print(A198631_list(36)) # Peter Luschny, Feb 21 2016

a(n) = numerator(sum(E(n,m),m=0..n)), n>=0, with the Euler triangle E(n,m)=A060096(n,m)/A060097(n,m).
E.g.f.: 2/(1+exp(-x)) (see a comment in  A060096).
r(n) := sum(E(n,m),m=0..n) = ((-1)^n)*sum(((-1)^m)*m!*S2(n,m)/2^m, m=0..n), n>=0, where S2 are the Stirling numbers of the second kind A048993. From the e.g.f. with y=exp(-x), dx=-y*dy, putting y=1 at the end. - Wolfdieter Lang, Nov 03 2011
a(n) = numerator(euler(n,1)/(2^n-1)) for n > 0. - Peter Luschny, Jul 14 2013
a(n) = numerator(2*(2^n-1)*B(n,1)/n) for n > 0, B(n,x) the Bernoulli polynomials. - Peter Luschny, May 24 2014
Numerators of the Taylor series coefficients 4*(2^(n+1)-1)*B(n+1)/(n+1) for n>0 of 1 + 2 * tanh(x/2) (cf. A000182 and A089171). - Tom Copeland, Oct 19 2016
a(n) = -2*zeta(-n)*A335956(n+1). - Peter Luschny, Jul 21 2020
Conjecture: r(n) = Sum_{k=0..n} A001147(k) * A039755(n, k) * (-1)^k / (k+1) where r(n) = a(n) / A006519(n+1) = (n!) * ([x^n] (2 / (1 + exp(-x)))), for n >= 0. - Werner Schulte, Feb 16 2024

New name, a simpler standalone definition by Peter Luschny, Jul 13 2012

Second comment corrected by Robert Israel, Feb 21 2016

A060096 Numerator of coefficients of Euler polynomials (rising powers).

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, 0, -3, 1, 0, 1, 0, -2, 1, -1, 0, 5, 0, -5, 1, 0, -3, 0, 5, 0, -3, 1, 17, 0, -21, 0, 35, 0, -7, 1, 0, 17, 0, -28, 0, 14, 0, -4, 1, -31, 0, 153, 0, -63, 0, 21, 0, -9, 1, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 691, 0, -1705, 0, 2805, 0, -231, 0, 165, 0, -11, 1, 0, 2073, 0, -3410, 0, 1683, 0, -396

Offset: 0

Wolfdieter Lang, Mar 29 2001

From S. Roman, The Umbral Calculus (see the reference in A048854), p. 101, (4.2.10) (corrected): E(n,x)= sum(sum(binomial(n,m)*((-1/2)^j)*j!*S2(n-m,j),j=0..k)*x^m,m=0..n), with S2(n,m)=A008277(n,m) and S2(n,0)=1 if n=0 else 0 (Stirling2).
From Wolfdieter Lang, Oct 31 2011: (Start)
This is the Sheffer triangle (2/(exp(x)+1),x) (which would be called in the above mentioned S. Roman reference Appell for (exp(t)+1)/2) (see p. 27).
  The e.g.f. for the row sums is 2/(1+exp(-x)). The row sums look like A198631(n)/A006519(n+1), n>=0.
  The e.g.f. for the alternating row sums is 2/(exp(x)*(exp(x)+1)). These sums look like (-1)^n*A143074(n)/ A006519(n+1).
  The e.g.f. for the a-sequence of this Sheffer array is 1. The z-sequence has e.g.f. (1-exp(x))/(2*x). This z-sequence is -1/(2*A000027(n))=-1/(2*(n+1)) (see the link under A006232 for the definition of a- and z-sequences). This leads to the recurrences given below.
The alternating power sums for the first n positive integers are given by sum((-1)^(n-j)*j^k,j=1..n) = (E(k, x=n+1)+(-1)^n*E(k, x=0))/2, k>=1, n>=1,with the row polynomials E(n, x)(see the Abramowitz-Stegun reference, p. 804, 23.1.4, and an addendum in the W. Lang link under A196837).
(End)

			n\m  0    1    2    3    4    5    6    7  8  ...
0:   1
1:  -1    1
2:   0   -1    1
3:   1    0   -3    1
4:   0    1    0   -2    1
5:  -1    0    5    0   -5    1
6:   0   -3    0    5    0   -3    1
7:  17    0  -21    0   35    0   -7    1
8:   0   17    0  -28    0   14    0   -4  1
...
The rational triangle a(n,m)/A060097(n,m) starts
n\m  0    1    2    3    4    5    6    7  8  ...
0:   1
1: -1/2   1
2:   0   -1    1
3:  1/4   0  -3/2   1
4:   0    1    0   -2    1
5: -1/2   0   5/2   0  -5/2   1
6:   0   -3    0    5    0   -3    1
7: 17/8   0 -21/2   0  35/4   0  -7/2   1
8:   0   17    0  -28    0   14    0   -4  1
...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 20, equations 20:4:1 - 20:4:8 at pages 177-178.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A060097.

A060096 := proc(n,m) coeff(euler(n,x),x,m) ; numer(%) ;end proc:
seq(seq(A060096(n,m),m=0..n),n=0..12) ; # R. J. Mathar, Dec 21 2010

Numerator[Flatten[Table[CoefficientList[EulerE[n, x], x], {n, 0, 12}]]] (* Jean-François Alcover, Apr 29 2011 *)

E(n, x)= sum((a(n, m)/b(n, m))*x^m, m=0..n), denominators b(n, m)= A060097(n, m).
From Wolfdieter Lang, Oct 31 2011: (Start)
E.g.f. for E(n, x) is 2*exp(x*z)/(exp(z)+1).
E.g.f. of column no. m, m>=0, is 2*x^{m+1}/(m!*(exp(x)+1)).
Recurrences for E(n,m):=a(n,m)/A060097(n,m) from the Sheffer a-and z-sequence:
E(n,m)=(n/m)*E(n-1,m-1), n>=1,m>=1.
E(n,0)=-n*sum(E(n-1,j)/(2*(j+1)),j=0..n-1), n>=1, E(0,0)=1.
(see the Sheffer comments above).
(End)
E(n,m) = binomial(n,m)*sum(((-1)^j)*j!*S2(n-m,j)/2^j ,j=0..n-m), 0<=m<=n, with S2 given by A008277. From S. Roman, The umbral calculus, reference under A048854, eq. (4.2.10), p. 101, with a=1, and a misprint corrected: replace 1/k! by binomial(n,k) (also in the two preceding formulas). - Wolfdieter Lang, Nov 03 2011
The first (m=0) column of the rational triangle is conjectured to be E(n,0) = ((-1)^n)*A198631(n) / A006519(n+1). See also the first column shown in A209308 (different signs). - Wolfdieter Lang, Jun 15 2015

Table rewritten by Wolfdieter Lang, Oct 31 2011

A208679 Kashaev's invariant for the (5,2)-torus knot (Solomon's seal knot).

Original entry on oeis.org

1, 71, 14641, 6242711, 4555133281, 5076970085351, 8024733763147921, 17074591123571719991, 47056485265721520250561, 163059403058191163396938631, 693897612604719894794535433201

Offset: 1

Peter Bala, Mar 01 2012

In general, Kashaev’s invariant for the (2*m+1,2)-torus knot has e.g.f. 1/2*sin(2*x)/cos((2*m+1)*x). Case m = 1 is A002439. For other examples see A208680 and A208681.
From Peter Bala, Dec 20 2021: (Start)
We make the following conjectures:
1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, ...] with an apparent pre-period of length 1 and a period [8, 7, 5, 1, 2, 4] of length 6 = phi(9).
2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients. (End)

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Cf. A002439 ((3,2)-torus knot), A208680, A208681, A208730, A208733, A057569.

Cf. A060096, A060097.

A208679 := proc(n) option remember; if n = 1 then 1; else (-4)^(n-1) - add((-25)^k*binomial(2*n-1,2*k)*procname(n-k),k=1..n) ; end if; end proc:
seq(A208679(n),n = 1..20) # Peter Bala, Dec 20 2021
A208679 := 5*10^(2*n-2)*(-1)^n*euler(2*n-1,3/10):
seq(A208679(n),n = 1..11); # Miles Wilson, Aug 05 2024

nmax = 20; Table[(CoefficientList[Series[1/2*Sin[2*x]/Cos[5*x], {x, 0, 2*nmax}], x] * Range[0, 2*nmax - 1]!)[[j]], {j, 2, 2*nmax + 1, 2}] (* Vaclav Kotesovec, Aug 30 2015 *)

my(x='x+O('x^30), v=Vec(serlaplace((1/2)*sin(2*x)/cos(5*x)))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, Aug 08 2024

E.g.f.: (1/2)*sin(2*x)/cos(5*x) = x + 71*x^3/3! + 14641*x^5/5! + ....
Define F(q) := Sum_{m,n >= 0} (q^(-m*n)*product {i = 1.. m+n} (1-q^i)). For the expansion of F(1-q) and F(exp(-t)) see A208733 and A208730 respectively. Kitami gives the conjectural e.g.f. exp(-9*t)*F(exp(-40*t)) = 1 + 71*t + 14641*t^2/2! + ....
a(n) = (-1)^n/(4*n+4)*20^(2*n+1)*Sum_{k = 1..20} X(k)*B(2*n+2,k/20), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 20 given by X(n) = 0 except for X(20*n+3) = X(20*n+17) = 1 and X(20*n+7) = X(20*n+13) = -1.
a(n) = 1/2*(-1)^(n+1)*L(-2*n-1,X) in terms of the associated L-series attached to the periodic arithmetical function X.
a(n) ~ (2*n-1)! * 2^(2*n-3/2) * 5^(2*n-1) * sqrt(5-sqrt(5)) / Pi^(2*n). - Vaclav Kotesovec, Aug 30 2015
From Peter Bala, May 11 2017: (Start)
Let X = 40*x. G.f. with offset 0: A(x) = 1 + 71*x + 14641*x^2 + ... = 1/(1 + 9*x - 2*X/(1 - 3*X/(1 + 9*x - 9*X/(1 - 11*X/(1 + 9*x - 21*X/(1 - 24*X/(1 + 9*x - ...))))))), where the sequence [2, 3, 9, 11, ..., n*(5*n - 1)/2, n*(5*n + 1)/2, ...] of unsigned coefficients in the partial numerators of the continued fraction is A057569.
A(x) = 1/(1 + 49*x - 3*X/(1 - 2*X/(1 + 49*x - 11*X/(1 - 9*X/(1 + 49*x - 24*X/(1 - 21*X/(1 + 49*x - 42*X/(1 - 38*X/(1 + 49*x - ...))))))))), where the sequence [3, 2, 11, 9, 24, 21, ...] of unsigned coefficients in the partial numerators of the continued fraction is obtained by swapping pairs of adjacent terms of A057569. Let B(x) = 1/(1 - 9*x)*A(x/(1 - 9*x)), that is, B(x) is the 9_th binomial transform of A(x). Then B(x/40) = 1 + 2*x + 10*x^2 + 104*x^3 + ... is the o.g.f. for A208730. (End)
From Peter Bala, Dec 20 2021: (Start)
a(1) = 1, a(n) = (-4)^(n-1) - Sum_{k = 1..n} (-25)^k*C(2*n-1,2*k)*a(n-k).
a(n) == 71^(n-1) ( mod (2^7)*3*(5^2) ). (End)
a(n) = 5*10^(2*n - 2)*(-1)^n*E(2*n - 1, 3/10), where E(n,x) is the n-th Euler polynomial in x (A060096/A060097). - Miles Wilson, Aug 05 2024

A233508 Numerators of the triangle of polynomial coefficients P(0,x)=1, 2P(n)=(1+x)((1+x)^(n-1)+x^(n-1)). Of the first array of A133135.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 3, 5, 1, 1, 5, 5, 5, 3, 1, 1, 3, 15, 10, 15, 7, 1, 1, 7, 21, 35, 35, 21, 4, 1, 1, 4, 14, 28, 35, 28, 14, 9, 1, 1, 9, 18, 42, 63, 63, 42, 18, 5, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 11, 1

Offset: 0

Paul Curtz, Dec 11 2013

Discovered via Euler polynomials A060096(n)/A060097(n).

The fractional sequence is 1, 1, 1, 1/2, 3/2, 1, 1/2, 3/2, 2, 1, 1/2, 2, 3, 5/2, 1,... =a(n)/b(n). There is a correspondant sequence for Bernoulli polynomials (*).

			1,
1, 1,
1, 3, 1,
1, 3, 2, 1,
1, 2, 3, 5, 1,
1, 5, 5, 5, 3, 1, etc.

Cf. (*) A193815.

p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t[n_, k_] := Coefficient[p[n], x, k] // Numerator; Table[t[n, k], {n, 0, 10 }, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

a(n) = reduced A133138(n)/A007395.

A178395 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the inverse Euler polynomial E^{-1}(n,x), 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 14, 28, 35, 28, 14, 4, 1, 1, 9, 18, 42, 63, 63, 42, 18, 9, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1, 1, 6, 33, 110, 495, 396, 462, 396, 495, 110, 33, 6, 1

Offset: 0

Paul Curtz, May 27 2010

The triangle of fractions A060096(n,m)/A060097(n,m) contains the coefficients of the Euler Polynomial E(n,x) in row n. The matrix inverse of this triangle is
   1;
  1/2,  1;
  1/2,  1,   1;
  1/2, 3/2, 3/2,  1;
  1/2,  2,   3,   2,   1;
  1/2, 5/2,  5,   5,  5/2,  1;
and defines inverse Euler polynomials E^{-1}(n,x) assuming that row n and column m contain the coefficient [x^m] E^{-1}(n,x). The column m=0 is 1 if n=0, otherwise 1/2.
The current triangle T(n,m) shows the numerator of [x^m] E^{-1}(n,x).
Numerators of exponential Riordan array [(1+exp(x))/2,x]. Central coefficients T(2n,n) are A088218. - Paul Barry, Sep 07 2010

			From _Paul Barry_, Sep 07 2010: (Start)
Triangle begins
  1;
  1,   1;
  1,   1,   1;
  1,   3,   3,   1;
  1,   2,   3,   2,   1;
  1,   5,   5,   5,   5,   1;
  1,   3,  15,  10,  15,   3,   1;
  1,   7,  21,  35,  35,  21,   7,   1;
  1,   4,  14,  28,  35,  28,  14,   4,   1;
  1,   9,  18,  42,  63,  63,  42,  18,   9,   1;
  1,   5,  45,  60, 105, 126, 105,  60,  45,   5,   1; (End)

T.-X. He, L. C. Hsu, P. J.-S. Shiue, The Sheffer group and the Riordan group, Discr. Appl. Math. 155 (2007) 1895-1909.

Cf. A178474 (denominators).
Cf. A060096, A060097.
Cf. A088218, A026741, A061041, A064038.

nm := 15 : eM := Matrix(nm,nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1,m+1] := coeff(euler(n,x),x,m) ; end do: for m from n+1 to nm-1 do eM[n+1,m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%d,", numer(eM[n,m])) ; end do: end do: # R. J. Mathar, Dec 21 2010

(* The function RiordanArray is defined in A256893. *)
rows = 13;
R = RiordanArray[(1 + E^#)/2&, #&, rows, True];
R // Flatten // Numerator (* Jean-François Alcover, Jul 20 2019 *)

T(n,k)=numerator((binomial(n,k)+binomial(0,n-k))/2);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print());

T(n,0) = 1.
T(n,m) = T(n,n-m).
T(n,1) = A026741(n).
T(n,2) = A064038(n) (numerators related to A061041).
Number triangle T(n,k) = [k<=n]*numerator((C(n,k) + C(0,n-k))/2). - Paul Barry, Sep 07 2010

A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 4, -6, 0, 1, 5, -10, 0, 5, 0, 6, -15, 0, 15, 0, -3, 7, -21, 0, 35, 0, -21, 0, 8, -28, 0, 70, 0, -84, 0, 17, 9, -36, 0, 126, 0, -252, 0, 153, 0, 10, -45, 0, 210, 0, -630, 0, 765, 0, -155, 11, -55, 0, 330, 0, -1386, 0, 2805, 0, -1705, 0, 12, -66, 0, 495

Offset: 1

Ralf Stephan, Sep 08 2004

The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.

			G(1,x) = 1
G(2,x) = 2*x - 1
G(3,x) = 3*x^2 - 3*x
G(4,x) = 4*x^3 - 6*x^2 + 1
G(5,x) = 5*x^4 - 10*x^3 + 5*x
G(6,x) = 6*x^5 - 15*x^4 + 15*x^2 - 3
G(7,x) = 7*x^6 - 21*x^5 + 35*x^3 - 21*x

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, pp. 573-574.

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

A001489(n) = G(2n, 0) = -G(2n, 1). Cf. A081733.

Cf. A060096/A060097, A027641/A027642.

p := proc(n,x) local j,k; add(binomial(n,k)*add(binomial(k,j)*2^j*bernoulli(j), j=0..k-1)*x^(n-k),k=0..n) end;
seq(print(sort(p(n,x))),n=1..8); # Peter Luschny, Jul 07 2009

g[n_, x_] := Sum[ k Binomial[n, k] EulerE[k-1, 0] x^(n-k), {k, 1, n}]; Table[ CoefficientList[g[n, x], x] // Reverse, {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2013, after Peter Luschny *)

G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)),n),y,x)

E.g.f.: Sum_{n >= 1} G(n, x)*t^n/n! = 2*t*e^(x*t)/(1 + e^t).
G(n, x) = Sum_{k=1..n} k*C(n, k)* Euler(k-1, 0)*x^(n-k). - Peter Luschny, Jul 13 2009
G(n, x) = n*Euler(n-1,x) = Sum_{k=0..n} binomial(n,k)*Bernoulli(k)*2*(1-2^k)*x^(n-k), with the Euler polynomials Euler(n,x) (see A060096/A060097) and Bernoulli numbers A027641/A027642. See the Graham et al. reference, pp. 573-574, Exercise 7.52. - Wolfdieter Lang, Mar 13 2017

A342314 T(n, k) = [x^k] 2^n*(Euler(n, x/2) + Euler(n, x)), where Euler(n, x) are the Euler polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

2, -2, 3, 0, -6, 5, 4, 0, -15, 9, 0, 24, 0, -36, 17, -32, 0, 100, 0, -85, 33, 0, -288, 0, 360, 0, -198, 65, 544, 0, -1680, 0, 1190, 0, -455, 129, 0, 6528, 0, -8064, 0, 3696, 0, -1032, 257, -15872, 0, 48960, 0, -34272, 0, 10920, 0, -2313, 513, 0, -238080, 0, 293760, 0, -133056, 0, 30960, 0, -5130, 1025

Offset: 0

Peter Luschny, Mar 19 2021

			Table starts:
                             [0] 2
                           [1] -2, 3
                         [2] 0, -6, 5
                       [3] 4, 0, -15, 9
                     [4] 0, 24, 0, -36, 17
                  [5] -32, 0, 100, 0, -85, 33
               [6] 0, -288, 0, 360, 0, -198, 65
           [7] 544, 0, -1680, 0, 1190, 0, -455, 129
         [8] 0, 6528, 0, -8064, 0, 3696, 0, -1032, 257
   [9] -15872, 0, 48960, 0, -34272, 0, 10920, 0, -2313, 513

Cf. A060096/A060097, A163982 (row sums).

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E := (n,x) -> 2^n*(euler(n, x/2) + euler(n, x));
seq(CoeffList(E(n, x)), n=0..9);

A342315 T(n, k) = [x^k] 2^n*(Euler(n, x) - Euler(n, x/2)), where Euler(n, x) are the Euler polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, -2, 3, 0, 0, -9, 7, 0, 8, 0, -28, 15, 0, 0, 60, 0, -75, 31, 0, -96, 0, 280, 0, -186, 63, 0, 0, -1008, 0, 1050, 0, -441, 127, 0, 2176, 0, -6272, 0, 3472, 0, -1016, 255, 0, 0, 29376, 0, -30240, 0, 10584, 0, -2295, 511, 0, -79360, 0, 228480, 0, -124992, 0, 30480, 0, -5110, 1023

Offset: 0

Peter Luschny, Mar 19 2021

			Table starts:
                            [0] 0
                           [1] 0, 1
                         [2] 0, -2, 3
                        [3] 0, 0, -9, 7
                     [4] 0, 8, 0, -28, 15
                   [5] 0, 0, 60, 0, -75, 31
                [6] 0, -96, 0, 280, 0, -186, 63
            [7] 0, 0, -1008, 0, 1050, 0, -441, 127
         [8] 0, 2176, 0, -6272, 0, 3472, 0, -1016, 255
      [9] 0, 0, 29376, 0, -30240, 0, 10584, 0, -2295, 511

Cf. A060096/A060097, A163747 (row sums).

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E := (n, x) -> 2^n*(euler(n, x) - euler(n, x/2));
0,seq(CoeffList(E(n, x)), n = 0..10);

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