_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 19 2008

Cf. A101479; columns: A121413, A136738, A136739A121417, A121422; diagonals: A136740, A121427, A136741; variants: A136730, A136733.

nonn,tabl,new

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, 4, 2, 1, 30, 9, 3, 1, 335, 69, 15, 4, 1, 4984, 769, 118, 22, 5, 1, 92652, 11346, 1317, 178, 30, 6, 1, 2065146, 208914, 19311, 1995, 250, 39, 7, 1, 53636520, 4613976, 352636, 29126, 2820, 335, 49, 8, 1, 1589752230, 118840164, 7722840, 528097, 41061

0,4

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,...;

(1,2,3), 4, 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;

(4,9,15,22), 30, 39,49,60,72,85,99,114,130,147,165,184,204,225,247,...;

(30,69,118,178,250), 335, 434,548,678,825,990,1174,1378,1603,1850,...;

(335,769,1317,1995,2820,3810), 4984, 6362,7965,9815,11935,14349,...;

(4984,11346,19311,29126,41061,55410,72492), 92652, 116262, 143722,...;

(92652,208914,352636,528097,740035,993678,1294776,1649634), 2065146,..;

(2065146,4613976,7722840,11476963,15971180,21310710,27611970,35003430,43626510),..;

where the rows are generated as follows.

Start row 0 with all 1's; from then on,

remove the first n+2 terms (shown in parenthesis) from row n

and then take partial sums to yield row n+1.

Note the second upper diagonal forms column 0 and equals A121413:

[1,1,4,30,335,4984,92652,2065146,53636520,1589752230,52926799310,...].

which equals column 3 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'.

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

Cf. A101479; columns: A121413, A136738, A136739; diagonals: A136740, A136741; variants: A136730, A136733.

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 19 2008

approved