A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

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

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

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

Adjacent sequences: A108995 A108996 A108997 * A108999 A109000 A109001

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 17 2005

Status

approved