OFFSET
0,8
COMMENTS
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)
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
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
LINKS
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.
FORMULA
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)
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
EXAMPLE
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)
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) {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 */
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Nov 05 2009
EXTENSIONS
Name clarified by Peter Luschny, May 25 2015
STATUS
approved