A177994 Triangle read by rows, A177990 * A070909.

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1

Offset: 0

Author

Gary W. Adamson, May 16 2010

Keywords

• nonn
• tabl

Comments

Row sums = A101881: (1, 2, 4, 5, 8, 9, 13, 14,...).

Double Riordan array ( 1/((1 - x)*(1 - x^2)); x*(1 - x^2), x/(1 - x^2) ) as defined in Davenport et al. The set 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, forms a group under matrix multiplication. Here g, f_1 and f_2 denote power series with constant term equal to 1. This is the array (( 1/((1 - x)*(1 - x^2)), 1/(1 - x) )) in the notation of the Bala link. - Peter Bala, Aug 26 2021

Examples

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 2, 1, 1;
3, 1, 2, 1, 1, 1;
4, 1, 3, 1, 2, 1, 1;
4, 1, 3, 1, 2, 1, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
...
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
• Peter Bala, Matrices with repeated columns - the generalised Appell groups
• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Crossrefs

Cf. A177991 = A070909 * A177990.

Cf. A101881.

Programs

Haskell

a177994 n k = a177994_tabl !! n !! k
a177994_row n = a177994_tabl !! n
a177994_tabl = [1] : [1,1] : map f a177994_tabl
               where f xs@(x:_) = (x + 1) : 1 : xs
-- Reinhard Zumkeller, Feb 20 2015

Formula

As infinite lower triangular matrices, A177990 * A070909