A119881 -id:A119881 - cp's OEIS frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1

Offset: 0

Peter Luschny, Dec 29 2008

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
(+) W_n(1) = T_n are the signed tangent numbers, see A009006.
(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.
(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.
(+) | W_n([n odd]) | the number of alternating permutations A000111.
(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008
The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009
From Peter Bala, Jun 10 2009: (Start)
The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as
... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.
Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function
... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.
The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).
In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].
The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is
sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.
(End)
The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009
The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:
gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009
Another version is at A119879. - Philippe Deléham, Oct 26 2013

			1
x
x^2  -1
x^3  -3x
x^4  -6x^2   +5
x^5 -10x^3  +25x
x^6 -15x^4  +75x^2  -61
x^7 -21x^5 +175x^3 -427x

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife polynomials and Euler numbers
Wikipedia, Bernoulli number
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.
W_n(k), k=0,1,...
W_0:  1,  1,  1,  1,   1,   1, ........ A000012
W_1:  0,  1,  2,  3,   4,   5, ........ A001477
W_2: -1,  0,  3,  8,  15,  24, ........ A067998
W_3:  0, -2,  2, 18,  52, 110, ........ A121670
W_4:  5,  0, -3, 32, 165, 480, ........
W_n(k), n=0,1,...
k=0:  1,  0, -1,  0,   5,   0, -61, ... A122045
k=1:  1,  1,  0, -2,   0,  16,   0, ... A155585
k=2:  1,  2,  3,  2,  -3,   2,  63, ... A119880
k=3:  1,  3,  8, 18,  32,  48, 128, ... A119881
k=4:  1,  4, 15, 52, 165, 484, ........         [Peter Luschny, Jul 07 2009]

w := proc(n,x) local v,k,pow,chen; pow := (a,b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1,4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1,4) *2^iquo(m,2)) end; add(add((-1)^v*binomial(k,v)*pow(v+x+1,n)*chen(k),v=0..k), k=0..n) end:
# Coefficients with zeros:
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t),t,16),t,i),x,i-n),n=0..i)), i=0..8);
# Recursion
W := proc(n,z) option remember; local k,p;
if n = 0 then 1 else p := irem(n+1,2);
z^n - p + add(`if`(irem(k,2)=1,0,
W(k,0)*binomial(n,k)*(power(z,n-k)-p)),k=2..n-1) fi end:
# Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* Jean-François Alcover, Oct 03 2011, after g.f. *)
sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* Jean-François Alcover, May 21 2013 *)
Flatten@Table[Binomial[n, 2k] EulerE[2k], {n, 0, 12}, {k, 0, n/2}](* Oliver Seipel, Jan 14 2025 *)

def A046978(k):
    if k % 4 == 0:
        return 0
    return (-1)**(k // 4)
def A153641_poly(n, x):
    return expand(add(2**(-(k // 2))*A046978(k+1)*add((-1)**v*binomial(k,v)*(v+x+1)**n for v in (0..k)) for k in (0..n)))
for n in (0..7): print(A153641_poly(n, x))  # Peter Luschny, Oct 24 2011

W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(k,v)*c_k*(x+v+1)^n where c_k = frac((-1)^(floor(k/4))/2^(floor(k/2))) [4 not div k] (Iverson notation).
From Peter Bala, Jun 10 2009: (Start)
E.g.f.: 2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....
W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).
Fourier series expansion for the generalized Bernoulli polynomials:
B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.
B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.
(End)
E.g.f.: exp(x*t) * sech(t). - Peter Luschny, Jul 07 2009
O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - Peter Bala, Mar 11 2012
Conjectural o.g.f.: Sum_{n >= 0} (1/2^((n-1)/2))*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - Peter Bala, Oct 03 2016

A119879 Exponential Riordan array (sech(x),x).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1, 0, 12465, 0, -5124, 0, 630, 0, -36, 0, 1, -50521, 0, 62325, 0, -12810, 0, 1050, 0, -45, 0, 1

Offset: 0

Paul Barry, May 26 2006

Row sums have e.g.f. exp(x)*sech(x) (signed version of A009006). Inverse of masked Pascal triangle A119467. Transforms the sequence with e.g.f. g(x) to the sequence with e.g.f. g(x)*sech(x).
Coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows). Another version in A153641. - Philippe Deléham, Oct 26 2013
Relations to Green functions and raising/creation and lowering/annihilation/destruction operators are presented in Hodges and Sukumar and in Copeland's discussion of this sequence and 2020 pdf. - Tom Copeland, Jul 24 2020

			Triangle begins:
     1;
     0,    1;
    -1,    0,     1;
     0,   -3,     0,   1;
     5,    0,    -6,   0,   1;
     0,   25,     0, -10,   0,   1;
   -61,    0,    75,   0, -15,   0,   1;
     0, -427,     0, 175,   0, -21,   0,  1;
  1385,    0, -1708,   0, 350,   0, -28,  0,  1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera, 2015.
Tom Copeland, Discussion of this sequence
Tom Copeland, SKipping over Dimensions, Juggling Zeros in the Matrix, 2020.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414.
Peter Luschny, Additive decompositions of classical numbers via the Swiss-Knife polynomials [_Tom Copeland_, Oct 20 2015]
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.

Row sums are A155585. - Johannes W. Meijer, Apr 20 2011
Rows reversed: A081658.
Cf. A109449, A153641, A162660. - Philippe Deléham, Oct 26 2013
Cf. A000182, A046802, A119467, A133314.
Cf. A019538, A028246, A090582, A123125, A130850, A248727.

T := (n,k) -> binomial(n,k)*2^(n-k)*euler(n-k,1/2): # Peter Luschny, Jan 25 2009

T[n_, k_] := Binomial[n, k] 2^(n-k) EulerE[n-k, 1/2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Peter Luschny *)

{T(n,k) = binomial(n,k)*2^(n-k)*(2/(n-k+1))*(subst(bernpol(n-k+1, x), x, 1/2) - 2^(n-k+1)*subst(bernpol(n-k+1, x), x, 1/4))};
for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 25 2019

@CachedFunction
def A119879_poly(n, x) :
    return 1 if n == 0  else add(A119879_poly(k, 0)*binomial(n, k)*(x^(n-k)-1+n%2) for k in range(n)[::2])
def A119879_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A119879_poly(n,x)).coeffs()  # Peter Luschny, Jul 16 2012
# Alternatively:

# uses[riordan_array from A256893]
riordan_array(sech(x), x, 9, exp=true) # Peter Luschny, Apr 19 2015

Number triangle whose k-th column has e.g.f. sech(x)*x^k/k!.
T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(1/2) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n,k)*p{k}(0)*((n mod 2) - 1 + x^(n-k)). - Peter Luschny, Jul 16 2012
E.g.f.: exp(x*z)/cosh(x). - Peter Luschny, Aug 01 2012
Sum_{k=0..n} T(n,k)*x^k = A122045(n), A155585(n), A119880(n), A119881(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 27 2013
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of this entry, A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020
Triangle equals P*((I + P^2)/2)^(-1), where P denotes Pascal's triangle A007318. - Peter Bala, Mar 07 2024

A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

Original entry on oeis.org

1, 1, 3, 18, 165, 2000, 29855, 527632, 10762857, 248811264, 6428081979, 183537694208, 5739195739277, 195059957567488, 7159662639822615, 282252719348582400, 11894243092571825745, 533554809104057434112, 25384473065818477067123, 1276688324194885747474432

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A062024, A119880, A119881, A155585, A302583, A302584, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cosh[x], {x, 0, n}], {n, 0, 19}]
Table[2^n EulerE[n, (n + 1)/2], {n, 0, 19}]

a(n) ~ n^n / cosh(1). - Vaclav Kotesovec, Jun 08 2019

A225116 a(n) = 3^n*A_{n, 1/3}(-1) where A_{n, k}(x) are the generalized Eulerian polynomials.

Original entry on oeis.org

1, 5, 24, 110, 480, 2000, 8064, 32240, 130560, 531200, 2095104, 8030720, 33546240, 156569600, 536838144, 243660800, 8589803520, 244224819200, 137438429184, -28539130347520, 2199021158400, 4960294141952000, 35184363700224, -1015283149035274240, 562949919866880

Offset: 0

Peter Luschny, Apr 29 2013

sign

Peter Luschny, Generalized Eulerian polynomials.

Cf. A155585(n) = 1^n*A_{n, 1/1}(-1), A119881(n) = 2^n*A_{n, 1/2}(-1).

EulerianPolynomial := proc(n,k,x) local j; if x = 1 then k^n*n! else (1-x)^(1+n)*(1+add(binomial(n,j)* polylog(-j,x)*k^j, j = 0..n)) fi end:
A225116 := n -> 3^n*EulerianPolynomial(n, 1/3, -1);
seq(round(evalf(A225116(i), 24)), i = 0..24);

Table[2^(t+1)*(Zeta[-t]*(1-2^(t+1))+(2^t-1)), {t,0,24}] (* Peter Luschny, Jul 20 2013 *)
Table[EulerE[n, 3] 2^n , {n, 0, 20}] (* Vladimir Reshetnikov, Oct 21 2015 *)

from mpmath import mp, polylog
mp.dps = 32; mp.pretty = True
def A225116(n): return 2^(1+n)*(3^n+add(binomial(n,j)*polylog(-j,-1) *3^(n-j) for j in (0..n)))
[int(A225116(n)) for n in (0..24)]

a(n) = 2^(1+n)*(3^n+sum_{j=0..n}(binomial(n,j)*Li_{-j}(-1)*3^(n-j))).

a(n) = 2^(t+1)*(zeta(-t)*(1-2^(t+1))+(2^t-1)). - Peter Luschny, Jul 20 2013

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, 0, 0, 2, 1, 5, -2, 3, 3, 1, 0, 0, 2, 8, 4, 1, -61, 16, -3, 18, 15, 5, 1, 0, 0, 2, 32, 52, 24, 6, 1, 1385, -272, 63, 48, 165, 110, 35, 7, 1, 0, 0, 2, 128, 484, 480, 198, 48, 8, 1, -50521, 7936, -1383, 528, 1395, 2000, 1085, 322, 63, 9, 1

Offset: 0

Peter Luschny, Dec 14 2014

This two-dimensional array of numbers can be seen as a generalization of the Euler secant and Euler tangent numbers (which are in their compressed and signless form A000364 resp. A000182 or interleaved in A000111). The cases n=0 and n=1 reduce to their expanded and signed forms A122045 and A155585. Moreover the columns are the values of the Swiss-Knife polynomials A153641 evaluated at the nonnegative integers.

Subsequences [3,3,1], [8,4,1], [15,5,1], [24,6,1], [35,7,1], [48,8,1], [63,9,1] found in rows of this entry, as a triangular array, are present in the antidiagonals of Table 5 of the East and Gray reference (A244490), and some subsequences in the rows of Table 5 are found in the antidiagonals of this entry, including [3,2,1] and [1,1]. Equivalently, the first four columns of Table 5 are embedded in this entry viewed as a square array on the table page. An explicit formula with combinatorial interpretations for these numbers is provided in the reference, and others are known for the corresponding columns for the modified Hermite polynomials of A244490. - Tom Copeland, Oct 04 2016

			Square array starts:
  [n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]     [8]
  [0]   1, 0, -1,   0,    5,    0,   -61,      0,   1385, ... A122045
  [1]   1, 1,  0,  -2,    0,   16,     0,   -272,      0, ... A155585
  [2]   1, 2,  3,   2,   -3,    2,    63,      2,  -1383, ... A119880
  [3]   1, 3,  8,  18,   32,   48,   128,    528,    512, ... A119881
  [4]   1, 4, 15,  52,  165,  484,  1395,   4372,  14505, ...
  [5]   1, 5, 24, 110,  480, 2000,  8064,  32240, 130560, ... A225116
  [6]   1, 6, 35, 198, 1085, 5766, 29855, 151878, 766745, ...
  A000012, A001477, A067998, A121670, ...
Triangular array starts:
                1,
              0,  1,
           -1,  1,  1,
          0,  0,  2,  1,
        5, -2,  3,  3,  1,
      0,  0,  2,  8,  4,  1,
  -61, 16, -3, 18, 15,  5,  1.

J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Cf. A122045, A155585, A119880, A119881, A225116, A000364, A000182, A000111, A067998, A121670, A153641, A247501.

Cf. A244490.

# EGF (row)
egf := n -> exp(n*x)*sech(x):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..8)), n=0..6);
# Swiss-Knife polynomial (column)
SKP := proc(n, x) local v, k, A; A := k -> `if`(k mod 4 = 0,0,(-1)^iquo(k,4)); add(2^iquo(-k,2)*A(k+1)*add((-1)^v* binomial(k,v)*(v+x+1)^n,v=0..k), k=0..n); expand(%) end:
seq(print(seq(SKP(k, n), n=0..9)), k=0..6);
# OGF (column)
col := proc(n, len) local T; T := A247501_row(n);
(-1)^(n+1)*add(T[k+1]/(x-1)^(k+1),k=0..n);
seq(coeff(series(%,x,len+1),x,j),j=0..len) end:
seq(print(col(n,8)), n=0..6);

nmax = 10; Clear[row]; row[n_] := row[n] = CoefficientList[Exp[n*x]*Sech[x] + O[x]^(nmax+2), x][[1 ;; nmax+1]]*Range[0, nmax]!;
rows = Table[row[n], {n, 0, nmax}];
T[n_, k_] := rows[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 03 2017 *)

G.f. for column k: the k-th column consists of the values of the k-th Swiss-Knife polynomial skp_{k}(x) evaluated at x = 0,1,2,...

O.g.f. for column k: Sum_{j=0..k} (-1)^(k+1)*A247501(k,j)/(x-1)^(j+1).

cp's OEIS Frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

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A119879 Exponential Riordan array (sech(x),x).

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A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

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A225116 a(n) = 3^n*A_{n, 1/3}(-1) where A_{n, k}(x) are the generalized Eulerian polynomials.

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A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k] exp(n*x)*sech(x), n>=0, k>=0.

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A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.