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Row n has length n+1.

From Gary W. Adamson, May 15 2010: (Start)

Eigensequence of the triangle = A038754 (i.e., 1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909.

Binomial transform of A070909 = triangle A177953. (End)

From Paul Barry, Nov 03 2010: (Start)

Generalized (conditional) Riordan array with k-th column generated by x^k/(1-x) if k is even, x^k otherwise.

A181651 is an eigentriangle. Inverse is A181650. (End)

From Peter Bala, Aug 15 2021: (Start)

Double Riordan array (1/(1 - x); x*(1 - x), x/(1 - x)) as defined in Davenport et al. The inverse array is the double Riordan array (1 - x - x^2; x/(1 - x - x^2), x*(1 - x - x^2)).

In general, double Riordan arrays of the form (g(x); x/g(x), x*g(x)), where g(x) = 1 + g_1*x + g_2*x^2 + ..., form a group under matrix multiplication with the group law given by (g(x); x/g(x), x*g(x)) * (G(x); x/G(x), x*G(x)) = (h(x); x/h(x), x*h(x)), where h(x) = G(x) + (g(x) - 1)*(G(x) + G(-x))/2. The inverse array of (g(x); x/g(x), x*g(x)) equals (f(x); x/f(x), x*f(x)), where f(x) = (2 - (g(x) - g(-x)))/(g(x) + g(-x)). (End)

Equals row sums of triangle A177994. - Gary W. Adamson, May 16 2010

From Ralf Stephan, Mar 09 2014: (Start)

Write the positive integers in a skewed triangle:

1, 2;

0, 3, 4, 5;

0, 0, 6, 7, 8, 9;

0, 0, 0, 10, 11, 12, 13, 14;

...

Sequence consists of the first number in each column. (End)

In a regular k-polygon draw lines connecting all the vertices. Select a triangle that tiles the polygon into k pieces. This triangle contains two adjacent polygon vertices. The third vertex is for even k the center of the polygon and for odd k one of the vertices of the central k-polygon (which is not included in the tiling). Count all lines connecting vertices in the original k-polygon that passes through the interior of the tiling triangle. That count is a(k-5). (See illustrations below.) - Lars Blomberg, Feb 20 2020

a(n) is the smallest number which has n+1 as a part in any of its maximally refined strict partitions. The first such are:(1),(2),(1,3),(1,4),(1,2,5),(1,2,6),(1,2,3,7),(1,2,3,8),(1,2,3,4,9) etc. - Sigurd Kittilsen, Oct 18 2024

Rows alternate between all 1's and alternating 1's and 0's. A 'mixed' sequence array: rows alternate between the rows of the sequence array for the all 1's sequence and the sequence array for the sequence 1,0,1,0,...

Column 2*k has g.f. x^(2*k)/(1-x); column 2*k+1 has g.f. x^(2*k+1)/(1-x^2).

Row sums are A029578(n+2). Antidiagonal sums are A106466.

This triangle is the Kronecker product of an infinite lower triangular matrix filled with 1's with a 2 X 2 lower triangular matrix of 1's. - Christopher Cormier, Sep 24 2017

From Peter Bala, Aug 21 2021: (Start)

Using the notation of Davenport et al.:

This is the double Riordan array ( 1/(1 - x); x/(1 + x), x*(1 + x) ).

The inverse array equals ( (1 - x)*(1 - x^2); x*(1 - x), x*(1 + x) ).

They are examples of double Riordan arrays of the form (g(x); x*f_1(x), x*f_2(x)), where f_1(x)*f_2(x) = 1. Arrays of this type form a group under matrix multiplication. For the group law see the Bala link. (End)

Row sums = A117142: (1, 2, 3, 5, 6, 9,...)

Previous name was: Signed version of Pascal's triangle.

Related to A000111 via its matrix inverse A162170.

For odd columns k, T(n, k) = binomial(n-1, k-1) * (-1)^floor((n+k-1)/2). For even columns, T(n, k) = 1 if n = k, otherwise 0. - Mike Tryczak, Jun 17 2015

From Peter Bala, Sep 08 2021: (Start)

In the notation of the Bala link, this is the array [[ cos(x) - sin(x), 1 ]] with inverse array A162170 = [[ sec(x) + tan(x), 1 ]].

In general, arrays of the form [[ G(x), 1 ]], where G(x) = 1 + g(1)*x + g(2)*x^2/2! + g(3)*x^3/3! + ... is an e.g.f., form a group with group law [[ G(x), 1 ]]*[[ F(x), 1 ]] = [[ G(x)*F_e(x) + F_o(x), 1 ]] and inverse array [[ G(x), 1 ]]^(-1) = [[ (1 - G_o(x))/G_e(x), 1 ]], where G_e(x) = (G(x) + G(-x))/2 and G_o(x) = (G(x) - G(-x))/2 are the even and odd parts of G(x). (End)