A136730 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.

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, 20, 6, 1, 224504, 28432, 3510, 527, 105, 27, 7, 1, 5051866, 523290, 50528, 5624, 795, 149, 35, 8, 1, 132523155, 11587072, 913377, 79553, 8396, 1140, 203, 44, 9, 1, 3969912160

Offset: 0

Author

Paul D. Hanna, Jan 19 2008

Keywords

• nonn
• tabl

Examples

			The square array begins:
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,...];
(1), 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
(2,5), 9, 14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,..;
(9,23,43), 70, 105,149,203,268,345,435,539,658,793,945,1115,1304,1513,.;
(70,175,324,527), 795, 1140,1575,2114,2772,3565,4510,5625,6929,8442,...;
(795,1935,3510,5624,8396), 11961, 16471,22096,29025,37467,47652,59832,..;
(11961,28432,50528,79553,117020,164672), 224504, 298786,390087,501300,..;
(224504,523290,913377,1414677,2050345,2847156,3835910), 5051866, 6535206,.;
(5051866,11587072,19918602,30410985,43486800,59633775,79412515,103464895),.;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note that the main diagonal forms column 0 and equals A101482:
[1,1,2,9,70,795,11961,224504,5051866,132523155,3969912160,...]
which equals column 1 of triangle A101479:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
where row n equals row (n-1) of T^(n-1) with appended '1'.
		

Crossrefs

Cf. A101479; columns: A101482, A136731, A136732; variants: A136733, A136737.

Programs

PARI

{T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n-1)))}