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A108998
Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.
5
1, 1, 0, 1, 2, 0, 1, 8, 2, 0, 1, 18, 16, 2, 0, 1, 32, 74, 24, 2, 0, 1, 50, 224, 170, 32, 2, 0, 1, 72, 530, 768, 306, 40, 2, 0, 1, 98, 1072, 2562, 1856, 482, 48, 2, 0, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 2, 0, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 2, 0
OFFSET
0,5
COMMENTS
Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.
LINKS
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
Cf. A108999 (main diagonal), A109000 (antidiagonal sums), A109001, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).
Sequence in context: A316649 A065329 A352772 * A356265 A309993 A248673
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 17 2005
STATUS
approved