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Triangle T(n,k) read by rows; related to number of preorders.
3

%I #27 Jan 18 2019 04:40:52

%S 1,0,2,0,3,6,0,0,20,24,0,0,15,130,120,0,0,0,210,924,720,0,0,0,105,

%T 2380,7308,5040,0,0,0,0,2520,26432,64224,40320,0,0,0,0,945,44100,

%U 303660,623376,362880,0,0,0,0,0,34650,705320,3678840,6636960,3628800

%N Triangle T(n,k) read by rows; related to number of preorders.

%C There are only m nonzero entries in the m-th column.

%H G. C. Greubel, <a href="/A079510/b079510.txt">Rows n=1..30 of triangle, flattened</a>

%H G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (See the array on page 29.)

%H G. Kreweras, <a href="/A019538/a019538.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)

%e Triangle begins:

%e 1;

%e 0, 2;

%e 0, 3, 6;

%e 0, 0, 20, 24;

%e 0, 0, 15, 130, 120;

%e ...

%t T[n_, k_]:= If[k < 1 || k > n, 0, If[n==1 && k==1, 1, n*(T[n-1, k-1] + T[n-2, k-1])]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* _G. C. Greubel_, Jan 17 2019 *)

%o (PARI) T(n,k)=if(k<=0 || k>n, 0, if(n==1 && k==1, 1, n*(T(n-1,k-1)+T(n-2,k-1))));

%Y A rearrangement of the triangle in A008306. - _Benoit Cloitre_, Jan 27 2003

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Jan 21 2003

%E Recurrence and more terms from _Michael Somos_, Jan 23 2003

%E Offset changed to 1 by _G. C. Greubel_, Jan 17 2019