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A185896 Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx. 9
1, 0, 2, 2, 0, 6, 0, 16, 0, 24, 16, 0, 120, 0, 120, 0, 272, 0, 960, 0, 720, 272, 0, 3696, 0, 8400, 0, 5040, 0, 7936, 0, 48384, 0, 80640, 0, 40320, 7936, 0, 168960, 0, 645120, 0, 846720, 0, 362880, 0, 353792, 0, 3256320, 0, 8951040, 0, 9676800, 0, 3628800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

DEFINITION

Define polynomials R(n,t) with t = tan(x) by

... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x).

The first few are

... R(0,t) = 1

... R(1,t) = 2*t

... R(2,t) = 2+6*t^2

... R(3,t) = 16*t+24*t^3.

This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar].

The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294.

COMBINATORIAL INTERPRETATION

A combinatorial interpretation for the polynomial R(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.

Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0.

For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0).

Let sc be the number of sign changes through a snake

... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}.

For example, the snake 0 4 -3 -1 -2 0  has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0):

... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc.

See the example section below for the cases n=1 and n=2.

PRODUCTION MATRIX

Define three arrays R, L, and S as

... R = superdiag[2,3,4,...]

... L = subdiag[1,2,3,...]

... S = diag[2,4,6,...]

with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle.

On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below.

The three arrays satisfy the commutation relations

... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L

and hence give a representation of the Lie algebra sl(2).

LINKS

Table of n, a(n) for n=0..54.

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.

A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A (2007) 463, 2401-2414 doi:10.1098/rspa.2007.0001

Michael E. Hoffman, Derivative polynomials, Euler polynomials,and associated integer sequences, Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R21.

Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.

M. J-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.

FORMULA

GENERATING FUNCTION

E.g.f.:

(1)... F(t,z) = 1/(cos(z)-t*sin(z))^2 = sum {n>=0} R(n,t)*z^n/n!

= 1 +  (2*t)*z + (2+6*t^2)*z^2/2! + (16*t+24*t^3)*z^3/3! + ....

The e.g.f. equals the square of the e.g.f. of A104035.

Continued fraction representation for the o.g.f:

(2)... F(t,z) = 1/(1-2*t*z - 2*(1+t^2)*z^2/(1-4*t*z -...- n*(n+1)*(1+t^2)

*z^2/(1-2*n*(n+1)*t*z -....

RECURRENCE RELATION

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

ROW POLYNOMIALS

The polynomials R(n,t) satisfy the recurrence relation

(4)... R(n+1,t) = d/dt{(1+t^2)*R(n,t)} with R(0,t) = 1.

Let D be the derivative operator d/dt and U = t, the shift operator.

(5)... R(n,t) = (D + DUU)^n 1

RELATION WITH OTHER SEQUENCES

A) Derivative Polynomials A155100

The polynomials (1+t^2)*R(n,t) are the  polynomials P_(n+2)(t) of

A155100.

B) Bernoulli Numbers A000367 and A002445

Put S(n,t) = R(n,i*t), where i = sqrt(-1). We have the definite integral

evaluation

(6)... int((1-t^2)*S(m,t)*S(n,t), t = -1..1) = (-1)^((m-n)/2)*2^(m+n+3)

*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.

C) Zigzag Numbers A000111

(7)... R_n(1) = A000828(n+1) = 2^n*A000111(n+1).

D) Eulerian Numbers A008292

The polynomials R(n,t) are related to the Eulerian polynomials A(n,t) via

(8)... R(n,t) = (t+i)^n*A(n+1,(t-i)/(t+i))

with the inverse identity

(9)... A(n+1,t) = (-i/2)^n*(1-t)^n*R(n,i*(1+t)/(1-t)),

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 and i = sqrt(-1).

E) Ordered set partitions A019538

(10)... R(n,t) = (-2*i)^n*T(n+1,x)/x,

where x = i/2*t-1/2 and T(n,x) is the n-th row po1ynomial of A019538;

F) Miscellaneous

Column 1 is the sequence of tangent numbers - see A000182.

A000670(n+1) = (-i/2)^n*R(n,3*i).

A004123(n+2) = 2*(-i/2)^n*R(n,5*i).

A080795(n+1) =(-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)). - Peter Bala, Aug 26 2011

--------------

T(n,k)=(-1)^(n+1)*(-1)^((n-k)/2)*sum_{j=k+1..n+1} j! *stirling2(n+1,j) *2^(n+1-j) *(-1)^(n+j-k) *binomial(j-1,k)), see A059419.

For n>1 and  0<i<=n

sum_{j=i+1..n+1}((1-(-1)^(j-i))/(2*(j-i))*(-1)^((n-j)/2)*T(n,j))=(n+1)*(-1)^((n-1-i)/2)*T(n-1,i).

[Leonid Bedratyuk, Aug 12 2012]

G.f.:  1/G(0,t,x), where G(k,t,x) =  1 - 2*t*x - 2*k*t*x - (1+t^2)*(k+2)*(k+1)*x^2/G(k+1,t,x) ; (continued fraction due T. J. Stieltjes). - Sergei N. Gladkovskii, Dec 27 2013

EXAMPLE

Table begins

n\k|.....0.....1.....2.....3.....4.....5.....6

==============================================

0..|.....1

1..|.....0.....2

2..|.....2.....0.....6

3..|.....0....16.....0....24

4..|....16.....0...120.....0...120

5..|.....0...272.....0...960.....0...720

6..|...272.....0..3696.....0..8400.....0..5040

..

Examples of recurrence relation

T(4,2) = 3*(T(3,1) + T(3,3)) = 3*(16 + 24) = 120;

T(6,4) = 5*(T(5,3) + T(5,5)) = 5*(960 + 720) = 8400.

Example of integral formula (6)

... int  ((1-t^2)*(16-120*t^2+120*t^4)*(272-3696*t^2+8400*t^4-5040*t^6),

t = -1..1) =  2830336/1365 = -2^13*Bernoulli(12).

Examples of sign change statistic sc on snakes of type (0,0)

= = = = = = = = = = = = = = = = = = = = = =

.....Snakes....# sign changes sc.......t^sc

= = = = = = = = = = = = = = = = = = = = = =

n=1

...0 1 -2 0...........1................t

...0 2 -1 0...........1................t

yields R(1,t) = 2*t;

n=2

...0 1 -2 3 0.........2................t^2

...0 1 -3 2 0.........2................t^2

...0 2 1 3 0..........0................1

...0 2 -1 3 0.........2................t^2

...0 2 -3 1 0.........2................t^2

...0 3 1 2 0..........0................1

...0 3 -1 2 0.........2................t^2

...0 3 -2 1 0.........2................t^2

yields

R(2,t) = 2+6*t^2.

MAPLE

R = proc(n) option remember;

if n=0 then RETURN(1);

else RETURN(expand(diff((u^2+1)*R(n-1), u))); fi;

end proc;

for n from 0 to 12 do

t1 := series(R(n), u, 20);

lprint(seriestolist(t1));

od:

PROG

(PARI) {T(n, k) = if( n<0 || k<0 || k>n, 0, if(n==k, n!, (k+1)*(T(n-1, k-1) + T(n-1, k+1))))};

CROSSREFS

Cf. A000182, A000364, A000828 (row sums), A008292, A008293, A008294, A104035, A155100.

Sequence in context: A221396 A221337 A157077 * A076256 A127467 A271708

Adjacent sequences:  A185893 A185894 A185895 * A185897 A185898 A185899

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Feb 07 2011

STATUS

approved

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Last modified December 7 05:39 EST 2016. Contains 278841 sequences.