OFFSET
0,5
COMMENTS
Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.
LINKS
Muniru A Asiru, Rows n=0..110 of antidiagonals, flattened
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
FORMULA
T(n, k) = Sum_{j=0..k} C(n+k-j-1, k-j)*(C(2*n+1, 2*j)-2*n*C(n-1, j-1)) for n >= k >= 0. G.f. for coordination sequence of B_n lattice: Sum(binomial(2*n+1, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]
EXAMPLE
Square array begins:
1, 0, 0, 0, 0, 0, 0, 0, ...
1, 2, 2, 2, 2, 2, 2, 2, ...
1, 8, 16, 24, 32, 40, 48, 56, ...
1, 18, 74, 170, 306, 482, 698, 954, ...
1, 32, 224, 768, 1856, 3680, 6432, 10304, ...
1, 50, 530, 2562, 8130, 20082, 42130, 78850, ...
1, 72, 1072, 6968, 28320, 85992, 214864, 467544, ...
1, 98, 1946, 16394, 83442, 307314, 907018, ...
Product of the g.f. of row n and (1-x)^n generates the rows of triangle A109001:
1;
1, 1;
1, 6, 1;
1, 15, 23, 1;
1, 28, 102, 60, 1;
1, 45, 290, 402, 125, 1;
1, 66, 655, 1596, 1167, 226, 1; ...
PROG
(PARI) T(n, k)=if(n<0 || k<0, 0, sum(j=0, k, binomial(n+k-j-1, k-j)*(binomial(2*n+1, 2*j)-2*n*binomial(n-1, j-1))))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 17 2005
STATUS
approved