0,5

Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.

R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1). Row sums are: 2^n*(2^n - n) for n>=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.]

G.f.s of initial rows of square array A108998 are:

(1),

(1 + x)/(1-x),

(1 + 6*x + x^2)/(1-x)^2;

(1 + 15*x + 23*x^2 + x^3)/(1-x)^3;

(1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.

Triangle begins:

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;

1,91,1281,4795,6155,2793,371,1;

1,120,2268,12040,23750,18888,5852,568,1;

1,153,3732,26628,74574,91118,49380,11124,825,1; ...

(PARI) T(n, k)=binomial(2*n+1, 2*k)-2*n*binomial(n-1, k-1)

Cf. A108998, A108999, A109000, 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: A146766 A146958 A154653 this_sequence A152602 A119726 A103999

Adjacent sequences: A108998 A108999 A109000 this_sequence A109002 A109003 A109004

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 17 2005