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A160185 Triangle read by rows, (1 / ((-1)*A129184 * A007318 + I)) - I, I = Identity matrix.

%I A160185 #24 Aug 03 2019 17:58:51
%S A160185 1,2,1,5,3,1,15,9,4,1,52,31,14,5,1,203,121,54,20,6,1,877,523,233,85,
%T A160185 27,7,1,4140,2469,1101,400,125,35,8,1,21147,12611,5625,2046,635,175,
%U A160185 44,9,1,115975,69161,30846,11226,3488,952,236,54,10,1
%N A160185 Triangle read by rows, (1 / ((-1)*A129184 * A007318 + I)) - I, I = Identity matrix.
%C A160185 Inverse binomial transform of the triangle shifts to left (= adding I as right border, I = Identity matrix); resulting in reversed rows of A121207.
%C A160185 Left border = Bell numbers, A000110 = eigensequence of Pascal's triangle.
%C A160185 Successive columns from left to right = eigensequences of Pascal's triangle deleting columns one at a time.
%C A160185 Row sums of the triangle = A060719: (1, 3, 9, 29, 103, ...). - _Gary W. Adamson_, May 20 2013
%C A160185 From _Gary W. Adamson_, Jul 18 2019: (Start)
%C A160185 Rows are eigensequences of triangles exemplified by the following arrangement of binomial sequences. Example: row 5 is (1, 5, 14, 31, 52, 0, 0, 0, ...), the eigensequence of:
%C A160185   1;
%C A160185   4, 1;
%C A160185   6, 3, 1;
%C A160185   4, 3, 2, 1;
%C A160185   1, 1, 1, 1, 1;
%C A160185   ... and the rest zeros.
%C A160185 Similarly, the production matrix for (1, 6, 20, 54, 121, 203, 0, 0, 0, ...) is:
%C A160185    1;
%C A160185    5, 1;
%C A160185   10, 4, 1;
%C A160185   10, 6, 3, 1;
%C A160185    5, 4, 3, 2, 1;
%C A160185    1, 1, 1, 1, 1, 1;
%C A160185   ... and the rest zeros. (End)
%F A160185 Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix.
%F A160185 Equals reversal by rows of triangle A121207, then delete right border. A121207 begins: 1; 1, 1; 1, 1, 2 1, 1, 3, 5; ...
%e A160185 First few rows of the triangle:
%e A160185        1;
%e A160185        2,     1;
%e A160185        5,     3,     1;
%e A160185       15,     9,     4,     1;
%e A160185       52,    31,    14,     5,    1;
%e A160185      203,   121,    54,    20,    6,   1;
%e A160185      877,   523,   233,    85,   27,   7,   1;
%e A160185     4140,  2469,  1101,   400,  125,  35,   8,  1;
%e A160185    21147, 12611,  5625,  2046,  635, 175,  44,  9,  1;
%e A160185   115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1;
%e A160185   ...
%Y A160185 Cf. A121207, A124496, A186020.
%Y A160185 Cf. A060719.
%K A160185 nonn,tabl
%O A160185 0,2
%A A160185 _Gary W. Adamson_, May 03 2009
%E A160185 Corrected by _Alois P. Heinz_, Apr 18 2013