cp's OEIS Frontend

(tan(x))^k = sum{n>0, If n+k is odd, T(n,k) = 0 = n!/k!*(-1)^((n+k)/2)*sum{j=k..n} (j!/n!) * Stirling2(n,j) * 2^(n-j) * (-1)^(n+j-k) * binomial(j-1,k-1)*x^n}. - Vladimir Kruchinin, Aug 13 2012

Also the Bell transform of A009006(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Number of divisors of 2n+2 of the form 2^k. - Giovanni Teofilatto, Jul 25 2007

Number of steps for iteration of map x -> (3/2)*ceiling(x) to reach an integer when started at 2*n+1.

Also number of steps for iteration of map x -> (3/2)*floor(x) to reach an integer when started at 2*n+3. - Benoit Cloitre, Sep 27 2003

The first time that a(n) = e+1 is when n is of the form 2^e - 1. - Robert G. Wilson v, Sep 28 2003

Let 2^k(n) = largest power of 2 dividing tangent number A000182(n). Then a(n-1) = 2*n - k(n). - Yasutoshi Kohmoto, Dec 23 2006

a(n) is the number of integers generated by b(i+1) = (3+2n)*(b(i) + b(i-1))/2, following these two initial values, b(0) = b(1) = 1. Thereafter only non-integers are generated. - Richard R. Forberg, Nov 09 2014

a(n) is the 2-adic valuation of 4*n+4, which is equal to the number of trailing 1-bits of 4*n+3 in binary. - Ruud H.G. van Tol, Sep 11 2023

Exponent of 2 in tangent numbers A000182.

Also, exponent of 2 in the sequences A008775, A009670, A009764, A009798, A012227, A024236, A024277, A024299, A052510.

Also, exponent of 2 in 4^(n-1)/n. [David Brink, Aug 08 2013]

Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x).

Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006

Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009

Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012

Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014

The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015

The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020

The n-th row of the triangle is obtained by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probabilities of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms of the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k-1) as the (2k-1)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k-1). - Luca Onnis, Oct 29 2023

Unsigned version is equal to A002425 up to n=11, but differs beyond that point.

Unsigned version: numerators of series coefficients of 1/(1 + cos(sqrt(x))); see Mathematica. - Clark Kimberling, Dec 06 2016

Note: Sum_{n>=0} (-1)^n*x^(3*n+1)/(3*n+1) = log( (1+x)/(1-x^3)^(1/3) )/2 + Pi*sqrt(3)/18 - atan( (1-2*x)*sqrt(3)/3 )*sqrt(3)/3.

The von Staudt-Clausen theorem states that this number is always an integer.

The denominators are given in A132050.

a(n)/n = 2, 2, 1, 4, 5, 32, 61, 544, ... are integers for n<=19. a(20)/20 = 58177770225664/5. - Paul Curtz, Mar 25 2013, Apr 04 2013

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).