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%I #2 Mar 30 2012 18:37:09
%S 1,1,1,2,2,1,9,5,3,1,70,23,9,4,1,795,175,43,14,5,1,11961,1935,324,70,
%T 20,6,1,224504,28432,3510,527,105,27,7,1,5051866,523290,50528,5624,
%U 795,149,35,8,1,132523155,11587072,913377,79553,8396,1140,203,44,9,1,3969912160
%N Square array, read by antidiagonals, where T(n,k) = T(n,k-1) + T(n-1,k+n-1) for n>0, k>0, such that T(n,0) = T(n-1,n-1) for n>0 with T(0,k)=1 for k>=0.
%e The square array begins:
%e 1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...];
%e (1), 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
%e (2,5), 9, 14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,..;
%e (9,23,43), 70, 105,149,203,268,345,435,539,658,793,945,1115,1304,1513,.;
%e (70,175,324,527), 795, 1140,1575,2114,2772,3565,4510,5625,6929,8442,...;
%e (795,1935,3510,5624,8396), 11961, 16471,22096,29025,37467,47652,59832,..;
%e (11961,28432,50528,79553,117020,164672), 224504, 298786,390087,501300,..;
%e (224504,523290,913377,1414677,2050345,2847156,3835910), 5051866, 6535206,.;
%e (5051866,11587072,19918602,30410985,43486800,59633775,79412515,103464895),.;
%e where the rows are generated as follows.
%e Start row 0 with all 1's; from then on,
%e remove the first n terms (shown in parenthesis) from row n
%e and then take partial sums to yield row n+1.
%e Note that the main diagonal forms column 0 and equals A101482:
%e [1,1,2,9,70,795,11961,224504,5051866,132523155,3969912160,...]
%e which equals column 1 of triangle A101479:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 3, 2, 1, 1;
%e 19, 9, 3, 1, 1;
%e 191, 70, 18, 4, 1, 1;
%e 2646, 795, 170, 30, 5, 1, 1;
%e 46737, 11961, 2220, 335, 45, 6, 1, 1;
%e 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
%e where row n equals row (n-1) of T^(n-1) with appended '1'.
%o (PARI) {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n-1)))}
%Y Cf. A101479; columns: A101482, A136731, A136732; variants: A136733, A136737.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Jan 19 2008
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