0,5

Row n equals the (n+1)-th differences of row n of the square array A108553. G.f. of row n equals: (1-x)^(n+1)*CBD_n(x), where CBD_n denotes the g.f. of the crystal ball sequence for D_n lattice.

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: (Start)

Let D_n be the root lattice generated as a monoid by {+-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(D_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(D_n) [Ardila et al.]. See A108556 for the corresponding array of f-vectors for these type D_n polytopes. See A008459 for the array of h-vectors for type A_n polytopes and A086645 for the array of h-vectors associated with type C_n polytopes.

The Hilbert transform of this array (as defined in A145905) equals A108553.

(End)

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

T(n, k) = C(2*n, 2*k) - 2*n*C(n-2, k-1) for n>1, with T(0, 0)=1, T(1, 0)=T(1, 1)=1. Row sums equal A008353: 2^(n-1)*(2^n-n) for n>1.

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: (Start)

O.g.f. : rational function N(x,z)/D(x,z), where N(x,z) = 1 - 3*(1 + x)*z + (3 + 2*x + 3*x^2)*z^2 - (1 + x)*(1 - 8*x + x^2)z^3 - 8*x*(1 + x^2)*z^4 + 2*x*(1 + x)*(1 - x)^2*z^5 and D(x,z) = ((1 - z)^2 - 2*x*z*(1 + z) + x^2*z^2)*(1 - z*(1 + x))^2.

For n >= 2, the row n generating polynomial equals 1/2*[(1 + sqrt(x))^(2n) + (1 - sqrt(x))^(2n)] - 2*n*x*(1 + x)^(n-2).

(End)

G.f.s of initial rows of square array A108553 are:

(1)/(1-x),

(1 + x)/(1-x)^2,

(1 + 2*x + x^2)/(1-x)^3,

(1 + 9*x + 9*x^2 + x^3)/(1-x)^4,

(1 + 20*x + 54*x^2 + 20*x^3 + x^4)/(1-x)^5,

(1 + 35*x + 180*x^2 + 180*x^3 + 35*x^4 + x^5)/(1-x)^6.

Triangle begins:

1;

1,1;

1,2,1;

1,9,9,1;

1,20,54,20,1;

1,35,180,180,35,1;

1,54,447,852,447,54,1;

1,77,931,2863,2863,931,77,1;

1,104,1724,7768,12550,7768,1724,104,1;

1,135,2934,18186,43128,43128,18186,2934,135,1;

1,170,4685,38200,124850,183356,124850,38200,4685,170,1; ...

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==0|n==1, 1, binomial(2*n, 2*k)-2*n*binomial(n-2, k-1)))}

Cf. A108553, A008353, A108558. Row n equals (n+1)-th differences of: A001844 (row 2), A005902 (row 3), A007204 (row 4), A008356 (row 5), A008358 (row 6), A008360 (row 7), A008362 (row 8), A008377 (row 9), A008379 (row 10).

A008459, A086645, A108556. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Sequence in context: A156883 A019803 A141601 this_sequence A128434 A119731 A155718

Adjacent sequences: A108555 A108556 A108557 this_sequence A108559 A108560 A108561

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2005