cp's OEIS Frontend

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

For a discussion and guide to related arrays, see A208510.

From Gus Wiseman, Jun 29 2025: (Start)

This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:

{1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5}

{2} {2,3} {2,3,4} {2,3,4,5}

{3} {3,4} {3,4,5}

{4} {4,5} {1,2,3,5}

{5} {1,2,4} {1,2,4,5}

{1,3} {1,2,5} {1,3,4,5}

{1,4} {1,3,4}

{1,5} {1,4,5}

{2,4} {2,3,5}

{2,5} {2,4,5}

{3,5}

{1,3,5}

For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)

To compute a(n): replace every other bit with zero (starting with the second bit) in each run of consecutive 1's in the binary expansion of n.

From Gus Wiseman, Jul 11 2025: (Start)

This is the greatest binary rank of a sparse subset of the binary indices of n, where:

1. The binary indices of a nonnegative integer are the positions of 1 in its reversed binary expansion.

2. A set is sparse iff 1 is not a first difference.

3. The binary rank of a set {S_1,S_2,...} is Sum_i 2^(S_i-1).

(End)

The equation "(k^2 - 8)/2 is a square" is a version of the generalized Pell Equation "x^2 - D*y^2 = C".

a(n)^2 - 2*A077444(n) = 8.

From Wolfdieter Lang, Jan 18 2013: (Start)

4*(1-z)/(1-6*z+z^2) = Sum_{n>=0} a(n+1)*z^n is the formal power series for tan(4*x)/tan(x) if one lets

z = (tan(x))^2. For the numerator and denominator of this o.g.f. see A034867 and A034839, respectively. Convergence holds for 0 <= z < 3 - 2*sqrt(2), approximately 0.1715728753. This means for |x| < Pi/8, approximately 0.3926990818.

See also the o.g.f. given by Johannes W. Meijer, Aug 01 2010, in the formula section of A001653 = (this sequence)/4.

(End)

Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 64 = 0. - Colin Barker, Feb 13 2014

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Skewed version of triangle in A200139.

Triangle without zeros: A207537.

For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014

This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015