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Triangle T(n,k) read by rows; given by [0,1,0,1,0,1,0,1,...] DELTA [1,1,1,2,1,3,1,4,1,5,1,6,...], where DELTA is Deléham's operator defined in A084938.
4

%I #20 Feb 19 2020 12:06:08

%S 1,0,1,0,1,2,0,1,5,5,0,1,9,22,15,0,1,14,60,98,52,0,1,20,130,366,457,

%T 203,0,1,27,245,1031,2190,2254,877,0,1,35,420,2436,7652,13251,11788,

%U 4140,0,1,44,672,5096,21862,55499,82288,65330,21147,0,1,54,1020,9744,54216,186595,402582,528400,382948,115975

%N Triangle T(n,k) read by rows; given by [0,1,0,1,0,1,0,1,...] DELTA [1,1,1,2,1,3,1,4,1,5,1,6,...], where DELTA is Deléham's operator defined in A084938.

%C T(n,k) appears to be the number of indecomposable permutations of [n+1] that avoid both of the dashed patterns 41-32 and 32-41 and contain n+1-k right-to-left minima. For example, T(3,1)=1 counts 4123 with 3 right-to-left minima; T(3,2)=5 counts 2413, 3142, 3412, 4213, 4312, each with 2 right-to-left minima; and T(3,3)=5 counts 2341, 2431, 3421, 4231, 4321, each with 1 right-to-left minimum. - _David Callan_, Aug 27 2014

%H Alois P. Heinz, <a href="/A085838/b085838.txt">Rows n = 0..140, flattened</a>

%F Sum_{k=0..n} (-x)^(n-k)*T(n,k) = A090365(n), A000110(n), A000012(n), A010892(n) for x=-1, 0, 1, 2. - _Philippe Deléham_, Oct 26 2006

%e 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 1, 5, 5;

%e 0, 1, 9, 22, 15;

%e 0, 1, 14, 60, 98, 52;

%e 0, 1, 20, 130, 366, 457, 203;

%e 0, 1, 27, 245, 1031, 2190, 2254, 877;

%e 0, 1, 35, 420, 2436, 7652, 13251, 11788, 4140;

%t m = 13;

%t (* DELTA is defined in A084938 *)

%t DELTA[Table[{0, 1}, {m/2 // Ceiling}] // Flatten, LinearRecurrence[{0, 2, 0, -1}, {1, 1, 1, 2}, m], m] // Flatten (* _Jean-François Alcover_, Feb 19 2020 *)

%Y Cf. A084938, A085791.

%Y Diagonals : A000007, A000012, A000096, A000110. Row sums : A090365

%K nonn,tabl

%O 0,6

%A _N. J. A. Sloane_, Aug 16 2003