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A321622 The Riordan square of the Fine numbers, triangle read by rows, T(n, k) for 0 <= k<= n.

%I A321622 #11 Mar 27 2020 13:59:34
%S A321622 1,1,1,0,1,1,1,1,1,1,2,4,2,1,1,6,10,7,3,1,1,18,31,19,10,4,1,1,57,97,
%T A321622 61,29,13,5,1,1,186,316,196,96,40,16,6,1,1,622,1054,652,316,136,52,19,
%U A321622 7,1,1,2120,3586,2210,1072,458,181,65,22,8,1,1
%N A321622 The Riordan square of the Fine numbers, triangle read by rows, T(n, k) for 0 <= k<= n.
%C A321622 Fine numbers as defined in A000957 have a(0) = 0 whereas our variant has a(0) = 1. The rows sums of the triangle are |A002420|.
%e A321622 [0] [    1]
%e A321622 [1] [    1,     1]
%e A321622 [2] [    0,     1,     1]
%e A321622 [3] [    1,     1,     1,     1]
%e A321622 [4] [    2,     4,     2,     1,     1]
%e A321622 [5] [    6,    10,     7,     3,     1,     1]
%e A321622 [6] [   18,    31,    19,    10,     4,     1,   1]
%e A321622 [7] [   57,    97,    61,    29,    13,     5,   1,  1]
%e A321622 [8] [  186,   316,   196,    96,    40,    16,   6,  1,  1]
%e A321622 [9] [  622,  1054,   652,   316,   136,    52,  19,  7,  1,  1]
%p A321622 # The function RiordanSquare is defined in A321620.
%p A321622 Fine := 1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)); RiordanSquare(Fine, 10);
%t A321622 (* The function RiordanSquare is defined in A321620. *)
%t A321622 FineGF = 1 + (1 - Sqrt[1 - 4x])/(3 - Sqrt[1 - 4x]);
%t A321622 RiordanSquare[FineGF, 10] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)
%o A321622 (Sage) # uses[riordan_square from A321620]
%o A321622 riordan_square(1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)), 10)
%Y A321622 T(n, 0) = A000957 (Fine), |A002420| (row sums), A000007 (alternating row sums).
%Y A321622 Cf. A321620.
%K A321622 nonn,tabl
%O A321622 0,11
%A A321622 _Peter Luschny_, Nov 22 2018