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

0,4

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

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).

Formula

As infinite lower triangular matrices, A177990 * A070909

Example

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;
...

Prog

(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

Crossrefs

Cf. A177991 = A070909 * A177990.

Cf. A101881.

Sequence in context: A034003 A306722 A168508 * A179285 A079211 A026835

Adjacent sequences: A177991 A177992 A177993 * A177995 A177996 A177997

Keyword

nonn,tabl

Author

Gary W. Adamson, May 16 2010

Status

approved