0,3

Row n equals the inverse binomial transform of row n of the square array A108553.

Array of f-vectors for type D root polytopes [Ardila et al.]. See A063007 and A127674 for the arrays of f-vectors for type A and type C root polytopes respectively. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

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]

Main diagonal equals A008353: 2^(n-1)*(2^n-n) for n>1.

O.g.f. : rational function N(x,z)/D(x,z), where N(x,z) = 1 - 3*(1 + 2*x)*z + (3 + 8*x + 8*x^2)*z^2 - (1 + 2*x)*(1 - 6*x - 6*x^2)z^3 - 8*x*(1 + x)(1 + 2*x + 2*x^2)*z^4 + 2*x*(1 + x)*(1 + 2*x)*z^5 and D(x,z) = ((1-z)^2 - 4*x*z)*(1 - z*(1 + 2*x))^2. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Triangle begins:

1;

1,2;

1,4,4;

1,12,30,20;

1,24,120,192,96;

1,40,330,940,1080,432;

1,60,732,3200,6240,5568,1856;

1,84,1414,8708,25200,37184,27104,7744;

1,112,2480,20352,80960,173824,206080,126976,31744; ...

(PARI) {T(n, k)=local(A=vector(n+1, r, vector(n+1, c, if(r-1==0|c-1==0, 1, if(r-1==1, 2*c-1, sum(j=0, c-1, binomial(r+c-j-2, c-j-1)*(binomial(2*r-2, 2*j)-2*(r-1)*binomial(r-3, j-1)))))))); polcoeff(subst(Ser(A[n+1]), x, x/(1+x))/(1+x), k)}

Cf. A108553, A108557 (row sums), A108558, Rows are inverse binomial transforms 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).

A063007, A127674. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Sequence in context: A134308 A117427 A097761 this_sequence A122440 A046943 A107728

Adjacent sequences: A108553 A108554 A108555 this_sequence A108557 A108558 A108559

nonn,tabl

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