A101343 -id:A101343 - cp's OEIS frontend

A155585 a(n) = 2^n*E(n, 1) where E(n, x) are the Euler polynomials.

1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0, 22368256, 0, -1903757312, 0, 209865342976, 0, -29088885112832, 0, 4951498053124096, 0, -1015423886506852352, 0, 246921480190207983616, 0, -70251601603943959887872, 0, 23119184187809597841473536, 0

Offset: 0

Paul D. Hanna, Jan 24 2009

Previous name was: a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1.
Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n > 0.
Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A155585(x) = Sum_{k=0..x-1} T(x,k)*(-1)^k. See Mathematica code below. - John M. Campbell, Nov 16 2011
For the difference table and the relation to the Seidel triangle see A239005. - Paul Curtz, Mar 06 2014
From Tom Copeland, Sep 29 2015: (Start)
Let z(t) = 2/(e^(2t)+1) = 1 + tanh(-t) = e.g.f.(-t) for this sequence = 1 - t + 2*t^3/3! - 16*t^5/5! + ... .
dlog(z(t))/dt = -z(-t), so the raising operators that generate Appell polynomials associated with this sequence, A081733, and its reciprocal, A119468, contain z(-d/dx) = e.g.f.(d/dx) as the differential operator component.
dz(t)/dt = z*(z-2), so the assorted relations to a Ricatti equation, the Eulerian numbers A008292, and the Bernoulli numbers in the Rzadkowski link hold.
From Michael Somos's formula below (drawing on the Edwards link), y(t,1)=1 and x(t,1) = (1-e^(2t))/(1+e^(2t)), giving z(t) = 1 + x(t,1). Compare this to the formulas in my list in A008292 (Sep 14 2014) with a=1 and b=-1,
A) A(t,1,-1) = A(t) = -x(t,1) = (e^(2t)-1)/(1+e^(2t)) = tanh(t) = t + -2*t^3/3! + 16*t^5/5! + -272*t^7/7! + ... = e.g.f.(t) - 1 (see A000182 and A000111)
B) Ainv(t) = log((1+t)/(1-t))/2 = tanh^(-1)(t) = t + t^3/3 + t^5/5 + ..., the compositional inverse of A(t)
C) dA/dt = (1-A^2), relating A(t) to a Weierstrass elliptic function
D) ((1-t^2)d/dt)^n t evaluated at t=0, a generator for the sequence A(t)
F) FGL(x,y)= (x+y)/(1+xy) = A(Ainv(x) + Ainv(y)), a related formal group law corresponding to the Lorentz FGL (Lorentz transformation--addition of parallel velocities in special relativity) and the Atiyah-Singer signature and the elliptic curve (1-t^2)*s = t^3 in Tate coordinates according to the Lenart and Zainoulline link and the Buchstaber and Bunkova link (pp. 35-37) in A008292.
A133437 maps the reciprocal odd natural numbers through the refined faces of associahedra to a(n).
A145271 links the differential relations to the geometry of flow maps, vector fields, and thereby formal group laws. See Mathworld for links of tanh to other geometries and statistics.
Since the a(n) are related to normalized values of the Bernoulli numbers and the Riemann zeta and Dirichlet eta functions, there are links to Witten's work on volumes of manifolds in two-dimensional quantum gauge theories and the Kervaire-Milnor formula for homotopy groups of hyperspheres (see my link below).
See A101343, A111593 and A059419 for this and the related generator (1 + t^2) d/dt and associated polynomials. (End)
With the exception of the first term (1), entries are the alternating sums of the rows of the Eulerian triangle, A008292. - Gregory Gerard Wojnar, Sep 29 2018

			E.g.f.: 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+ ... = exp(x)/cosh(x).
O.g.f.: 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 - 353792*x^11 +- ...
O.g.f.: 1 + x/(1+2*x) + 2!*x^2/((1+2*x)*(1+4*x)) + 3!*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. 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
Tom Copeland, The Kervaire-Milnor formula
H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 393-422. see section 12, pp. 410-411.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
G. Rzadkowski, Bernoulli numbers and solitons-revisited, Journal of Nonlinear Math. Physics, 1711, pp. 121-126.

Cf. A001057, A009006, A102573.
Equals row sums of A119879. - Johannes W. Meijer, Apr 20 2011
Cf. A081733, A119468, A008292, A000182, A000111, A101343, A111593, A059419, A133437, A145271.
(-1)^n*a(n) are the alternating row sums of A123125. - Wolfdieter Lang, Jul 12 2017

A155585 := n -> 2^n*euler(n, 1): # Peter Luschny, Jan 26 2009
a := proc(n) option remember; `if`(n::even, 0^n, -(-1)^n - add((-1)^k*binomial(n,k) *a(n-k), k = 1..n-1)) end: # Peter Luschny, Jun 01 2016
# Or via the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> (-2)^n*subs(x = -1/2, F(n)):
seq(a(n), n = 0..30); # Peter Luschny, May 21 2021

a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] (* Peter Luschny, Apr 29 2009 *)
poly[q_] :=  2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] :=  First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Table[Sum[T[x, k]*(-1)^k, {k, 0, x-1}], {x, 1, 16}] (* John M. Campbell, Nov 16 2011 *)
f[n_] := (-1)^n 2^(n+1) PolyLog[-n, -1]; f[0] = -f[0]; Array[f, 27, 0] (* Robert G. Wilson v, Jun 28 2012 *)

a(n)=if(n==0,1,sum(k=0,n-1,(-1)^(k)*binomial(n-1,k)*a(n-1-k)*a(k)))

a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp(X)/cosh(X),n)

a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1+2*k*x+x*O(x^n))),n) \\ Paul D. Hanna, Jul 20 2011

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( 1 + sinh(x + A) / cosh(x + A), n))} /* Michael Somos, Jan 16 2012 */

a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,intformal(subst(A,x,-x)+x*O(x^n))^k/k!));n!*polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 25 2013

from sympy import bernoulli
def A155585(n): return (((2<<(m:=n+1))-2)*bernoulli(m)<>1) if n&1 else (0 if n else 1) # Chai Wah Wu, Apr 14 2023

def A155585(n) :
    if n == 0 : return 1
    return add(add((-1)^(j+1)*binomial(n+1,k-j)*j^n for j in (0..k)) for k in (1..n))
[A155585(n) for n in (0..26)] # Peter Luschny, Jul 23 2012

def A155585_list(n): # Akiyama-Tanigawa algorithm
    A = [0]*(n+1); R = []
    for m in range(n+1) :
        d = divmod(m+3, 4)
        A[m] = 0 if d[1] == 0 else (-1)^d[0]/2^(m//2)
        for j in range(m, 0, -1) :
            A[j - 1] = j * (A[j - 1] - A[j])
        R.append(A[0])
    return R
A155585_list(30) # Peter Luschny, Mar 09 2014

E.g.f.: exp(x)*sech(x) = exp(x)/cosh(x). (See A009006.) - Paul Barry, Mar 15 2006
Sequence of absolute values is A009006 (e.g.f. 1+tan(x)).
O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 2*k*x). - Paul D. Hanna, Jul 20 2011
a(n) = 2^n*E_{n}(1) where E_{n}(x) are the Euler polynomials. - Peter Luschny, Jan 26 2009
a(n) = EL_{n}(-1) where EL_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
a(n+1) = (4^n-2^n)*B_n(1)/n, where B_{n}(x) are the Bernoulli polynomials (B_n(1) = B_n for n <> 1). - Peter Luschny, Apr 22 2009
G.f.: 1/(1-x+x^2/(1-x+4*x^2/(1-x+9*x^2/(1-x+16*x^2/(1-...))))) (continued fraction). - Paul Barry, Mar 30 2010
G.f.: -log(x/(exp(x)-1))/x = Sum_{n>=0} a(n)*x^n/(2^(n+1)*(2^(n+1)-1)*n!). - Vladimir Kruchinin, Nov 05 2011
E.g.f.: exp(x)/cosh(x) = 2/(1+exp(-2*x)) = 2/(G(0) + 1); G(k) = 1 - 2*x/(2*k + 1 - x*(2*k+1)/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2011
E.g.f. is x(t,1) + y(t,1) where x(t,a) and y(t,a) satisfy y(t,a)^2 = (a^2 - x(t,a)^2) / (1 - a^2 * x(t,a)^2) and dx(t,a) / dt = y(t,a) * (1 - a * x(t,a)^2) and are the elliptic functions of Edwards. - Michael Somos, Jan 16 2012
E.g.f.: 1/(1 - x/(1+x/(1 - x/(3+x/(1 - x/(5+x/(1 - x/(7+x/(1 - x/(9+x/(1 +...))))))))))), a continued fraction. - Paul D. Hanna, Feb 11 2012
E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(-x) dx )^n / n!. - Paul D. Hanna, Nov 25 2013
a(n) = -2^(n+1)*Li_{-n}(-1). - Peter Luschny, Jun 28 2012
a(n) = Sum_{k=1..n} Sum_{j=0..k} (-1)^(j+1)*binomial(n+1,k-j)*j^n for n > 0. - Peter Luschny, Jul 23 2012
From Sergei N. Gladkovskii, Oct 25 2012 to Dec 16 2013: (Start)
Continued fractions:
G.f.: 1 + x/T(0) where T(k) = 1 + (k+1)*(k+2)*x^2/T(k+1).
E.g.f.: exp(x)/cosh(x) = 1 + x/S(0) where S(k) = (2*k+1) + x^2/S(k+1).
E.g.f.: 1 + x/(U(0)+x) where U(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1)))).
E.g.f.: 1 + tanh(x) = 4*x/(G(0)+2*x) where G(k) = 1 - (k+1)/(1 - 2*x/(2*x + (k+1)^2/G(k+1)));
G.f.: 1 + x/G(0) where G(k) = 1 + 2*x^2*(2*k+1)^2 - x^4*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1) (due to Stieltjes).
E.g.f.: 1 + x/(G(0) + x) where G(k) = 1 - 2*x/(1 + (k+1)/G(k+1)).
G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*(k+1)/( 1 - x*(k+1)/Q(k+1)).
G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*k^2 + x/(1 - x*(k+1)^2/Q(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 - 2*x + x*(k+1)/(1-x*(k+1)/Q(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 - x*(k+1)/(1 + x*(k+1)/Q(k+1)).
E.g.f.: 1 + x*Q(0) where Q(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/Q(k+1)).
G.f.: 2 - T(0)/(1+x) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1+x)^2/T(k+1)).
E.g.f.: 1/(x - Q(0)) where Q(k) = 4*k^2 - 1 + 2*x + x^2*(2*k-1)*(2*k+3)/Q(k+1). (End)
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A001057. - Michael Somos, Jan 03 2013
From Paul Curtz, Mar 06 2014: (Start)
a(2n) = A000007(n).
a(2n+1) = (-1)^n*A000182(n+1).
a(n) is the binomial transform of A122045(n).
a(n) is the row sum of A081658. For fractional Euler numbers see A238800.
a(n) + A122045(n) = 2, 1, -1, -2, 5, 16, ... = -A163982(n).
a(n) - A122045(n) = -A163747(n).
a(n) is the Akiyama-Tanigawa transform applied to 1, 0, -1/2, -1/2, -1/4, 0, ... = A046978(n+3)/A016116(n). (End)
a(n) = 2^(2*n+1)*(zeta(-n,1/2) - zeta(-n, 1)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) = 2^(n + 1)*(2^(n + 1) - 1)*Bernoulli(n + 1, 1)/(n + 1). (From Bill Gosper, Oct 28 2015) - N. J. A. Sloane, Oct 28 2015 [See the above comment from Peter Luschny, Apr 22 2009.]
a(n) = -(n mod 2)*((-1)^n + Sum_{k=1..n-1} (-1)^k*C(n,k)*a(n-k)) for n >= 1. - Peter Luschny, Jun 01 2016
a(n) = (-2)^n*F_{n}(-1/2), where F_{n}(x) is the Fubini polynomial. - Peter Luschny, May 21 2021

New name from Peter Luschny, Mar 12 2015

A155100 Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 2, 2, 0, 8, 0, 6, 0, 16, 0, 40, 0, 24, 16, 0, 136, 0, 240, 0, 120, 0, 272, 0, 1232, 0, 1680, 0, 720, 272, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 7936, 0, 176896, 0, 814080, 0, 1491840

Offset: 0

N. J. A. Sloane, Nov 05 2009

The definition is d^(n-1) tan x / dx^n = P_n(tan x) for n>=1 and 1 for n=0.
Interpolates between factorials and tangent numbers.
From Peter Bala, Mar 02 2011: (Start)
Companion triangles are A104035 and A185896.
A combinatorial interpretation for the polynomial P_n(t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...,|x_n|} = {1,2,...,n}.
They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation and put x_0 = -(n+1) and x_(n+1) = (-1)^n*(n+1). Then x_0,x_1,...,x_n,x_(n+1) is a snake of type S(n) when x_0 < x_1 > x_2 < ... x_(n+1). For example, -5 4 -3 -1 -2 5 is a snake of type S(4).
Let sc be the number of sign changes through a snake sc = #{i, 0 <= i <= n, x_i*x_(i+1) < 0}. For example, the snake -5 4 -3 -1 -2 5 has sc = 3.
The polynomial P_(n+1)(t) is the generating function for the sign change statistic on snakes of type S(n): P_(n+1)(t) = sum {snakes in S(n)} t^sc.
See the example section below for the cases n=1 and n=2.
(End)
Equals A107729 when the first column is removed. - Georg Fischer, Jul 26 2023

			The polynomials P_{-1}(u) through P_6(u) with exponents in decreasing order:
      1
      u
      u^2 +    1
    2*u^3 +    2*u
    6*u^4 +    8*u^2 +    2
   24*u^5 +   40*u^3 +   16*u
  120*u^6 +  240*u^4 +  136*u^2 +  16
  720*u^7 + 1680*u^5 + 1232*u^3 + 272*u
  ...
Triangle begins:
  1
  0, 1
  1, 0, 1
  0, 2, 0, 2
  2, 0, 8, 0, 6
  0, 16, 0, 40, 0, 24
  16, 0, 136, 0, 240, 0, 120
  0, 272, 0, 1232, 0, 1680, 0, 720
  272, 0, 3968, 0, 12096, 0, 13440, 0, 5040
  0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320
  7936, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880
  0, 353792, 0, 3610112, 0, 12207360, 0, 18627840, 0, 13305600, 0, 3628800
  ...
From _Peter Bala_, Feb 07 2011: (Start)
Examples of sign change statistic sc on snakes of type S(n):
    Snakes     # sign changes sc  t^sc
  ===========  =================  ====
n=1:
  -2  1 -2 ........... 2 ........ t^2
  -2 -1 -2 ........... 0 ........ 1
                  yields P_2(t) = 1 + t^2;
n=2:
  -3  1 -2  3 ........ 3 ........ t^3
  -3  2  1  3 ........ 1 ........ t
  -3  2 -1  3 ........ 3 ........ t^3
  -3 -1 -2  3 ........ 1 ........ t
                  yields P_3(t) = 2*t + 2*t^3. (End)

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.

G. C. Greubel, Rows n = 0..101 of triangle, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374 [math.CO], 2021.

For other versions of this triangle see A008293, A101343.
A104035 is a companion triangle.
Highest order coefficients give factorials A000142. Constant terms give tangent numbers A000182. Other coefficients: A002301.
Setting u=1 in P_n gives A000831, u=2 gives A156073, u=3 gives A156075, u=4 gives A156076, u=1/2 gives A156102.
Setting u=sqrt(2) in P_n gives A156108 and A156122; setting u=sqrt(3) gives A156103 and A000436.
Cf. A008292, A019538, A107729, A185896.

P:=proc(n) option remember;
if n=-1 then RETURN(1); elif n=0 then RETURN(u); else RETURN(expand((u^2+1)*diff(P(n-1),u))); fi;
end;
for n from -1 to 12 do t1:=series(P(n),u,20); lprint(seriestolist(t1)); od:
# Alternatively:
with(PolynomialTools): seq(print(CoefficientList(`if`(i=0,1,D@@(i-1))(tan),tan)), i=0..7); # Peter Luschny, May 19 2015

p[n_, u_] := D[Tan[x], {x, n}] /. Tan[x] -> u /. Sec[x] -> Sqrt[1 + u^2] // Expand; p[-1, u_] = 1; Flatten[ Table[ CoefficientList[ p[n, u], u], {n, -1, 9}]] (* Jean-François Alcover, Jun 28 2012 *)
T[ n_, k_] := Which[n<0, Boole[n==-1 && k==0], n==0, Boole[k==1], True, (k-1)*T[n-1, k-1] + (k+1)*T[n-1, k+1]]; (* Michael Somos, Jul 09 2024 *)

{T(n, k) = if(n<0, n==-1 && k==0, n==0, k==1, (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1))}; /* Michael Somos, Jul 09 2024 */

If the polynomials are denoted by P_n(u), we have the recurrence P_{-1}=1, P_0 = u, P_n = (u^2+1)*dP_{n-1}/du.
G.f.: Sum_{n >= 0} P_n(u) t^n/n! = (sin t + u*cos t)/(cos t - u sin t). [Hoffman]
From Peter Bala, Feb 07 2011: (Start)
RELATION WITH BERNOULLI NUMBERS A000367 AND A002445
Put T(n,t) = P_n(i*t), where i = sqrt(-1). We have the definite integral evaluation, valid when both m and n are >=1 and m+n >= 4:
int( T(m,t)*T(n,t)/(1-t^2), t = -1..1) = (-1)^((m-n)/2)*2^(m+n-1)*Bernoulli(m+n-2).
The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
RELATION WITH OTHER ROW POLYNOMIALS
The following three identities hold for n >= 1:
P_(n+1)(t) = (1+t^2)*R(n-1,t) where R(n,t) is the n-th row polynomial of A185896.
P_(n+1)(t) = (-2*i)^n*(t-i)*R(n,-1/2+1/2*i*t), where i = sqrt(-1) and R(n,x) is an ordered Bell polynomial, that is, the n-th row polynomial of A019538.
P_(n+1)(t) = (t-i)*(t+i)^n*A(n,(t-i)/(t+i)), where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials - see A008292. (End)
T(n,k) = cos((n+k)*Pi/2) * Sum_{p=0..n-1} A008292(n-1,p+1) Sum_{j=0..k}(-1)^(p+j+1) * binomial(p+1,k-j) *binomial(n-p-1,j) for n>1. - Ammar Khatab, Aug 15 2024

Name clarified by Peter Luschny, May 25 2015

A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 8, 6, 16, 40, 24, 16, 136, 240, 120, 272, 1232, 1680, 720, 272, 3968, 12096, 13440, 5040, 7936, 56320, 129024, 120960, 40320, 7936, 176896, 814080, 1491840, 1209600, 362880, 353792, 3610112, 12207360, 18627840, 13305600, 3628800

Offset: 0

N. J. A. Sloane

James Gregory calculated the first eight rows of this table (with some numerical errors) in 1671. See Roy, p. 299. - Peter Bala, Sep 06 2016

			From _Peter Bala_, Sep 06 2016: (Start)
Table begins
    1
    1     1
    2     2
    2     8      6
   16    40     24
   16   136    240    120
  272  1232   1680    720
  272  3968  12096  13440  5040
  ...
D(tan(x)) = 1 + tan(x)^2.
D^2(tan(x)) = 2*tan(x) + 2*tan(x)^3.
D^3(tan(x)) = 2 + 8*tan(x)^2 + 6*tan(x)^4.
D^4(tan(x)) = 16*tan(x) + 40*tan(x)^3 + 24*tan(x)^5. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..990
William Y. C. Chen and Amy M. Fu, The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials, arXiv:2204.01497 [math.CO], 2022.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
R. Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Mathematics Magazine Vol. 63, No. 5 (Dec., 1990), 291-306.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

Cf. A008294. Other versions of same triangle: A101343, A155100.
T(n,ceiling(n/2)) gives A000142.
Bisection of column k=0 gives A000182.
Row sums give A000831.

row[n_] := CoefficientList[ D[Tan[x], {x, n}] /. Tan -> Identity /. Sec -> Function[Sqrt[1 + #^2]], x] // DeleteCases[#, 0]&; Table[row[n], {n, 0, 10}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Apr 05 2013 *)
T[ n_, k_] := If[n<1, Boole[n==0 && k==1], (k-1)*T[n-1, k-1] + (k+1)*T[n-1, k+1]]; (* Michael Somos, Jul 08 2024 *)

{T(n, k) = if(n<1, n==0 && k==1, (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1))}; /* Michael Somos, Jul 08 2024 */

T(0, k) = delta(1, k), T(n, k) = (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1).

cp's OEIS Frontend

A155585 a(n) = 2^n*E(n, 1) where E(n, x) are the Euler polynomials.

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A155100 Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0.

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A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.

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