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A177994 Triangle read by rows, A177990 * A070909.

%I A177994 #15 Jun 02 2025 02:56:45
%S A177994 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,
%T A177994 1,1,5,1,4,1,3,1,2,1,1,5,1,4,1,3,1,2,1,1,1
%N A177994 Triangle read by rows, A177990 * A070909.
%C A177994 Row sums = A101881: (1, 2, 4, 5, 8, 9, 13, 14,...).
%C A177994 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
%H A177994 Reinhard Zumkeller, <a href="/A177994/b177994.txt">Rows n = 0..100 of triangle, flattened</a>
%H A177994 Peter Bala, <a href="/A177994/a177994.pdf">Matrices with repeated columns - the generalised Appell groups</a>
%H A177994 D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="https://doi.org/10.37236/2034">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012).
%F A177994 As infinite lower triangular matrices, A177990 * A070909
%e A177994 First few rows of the triangle =
%e A177994 1;
%e A177994 1, 1;
%e A177994 2, 1, 1;
%e A177994 2, 1, 1, 1;
%e A177994 3, 1, 2, 1, 1;
%e A177994 3, 1, 2, 1, 1, 1;
%e A177994 4, 1, 3, 1, 2, 1, 1;
%e A177994 4, 1, 3, 1, 2, 1, 1, 1;
%e A177994 5, 1, 4, 1, 3, 1, 2, 1, 1;
%e A177994 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
%e A177994 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
%e A177994 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
%e A177994 7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
%e A177994 7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
%e A177994 ...
%o A177994 (Haskell)
%o A177994 a177994 n k = a177994_tabl !! n !! k
%o A177994 a177994_row n = a177994_tabl !! n
%o A177994 a177994_tabl = [1] : [1,1] : map f a177994_tabl
%o A177994                where f xs@(x:_) = (x + 1) : 1 : xs
%o A177994 -- _Reinhard Zumkeller_, Feb 20 2015
%Y A177994 Cf. A177991 = A070909 * A177990.
%Y A177994 Cf. A101881.
%K A177994 nonn,tabl
%O A177994 0,4
%A A177994 _Gary W. Adamson_, May 16 2010