A155100 - cp's OEIS frontend

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.

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

cp's OEIS Frontend

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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