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. - Peter Bala, Oct 23 2008

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 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. - Peter Bala, 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; ...

T[n_, k_] := Module[{A}, A = Table[Table[If[r - 1 == 0 || c - 1 == 0, 1, If[r - 1 == 1, 2c - 1, Sum[Binomial[r + c - j - 2, c - j - 1] (Binomial[2r - 2, 2j] - 2(r - 1) Binomial[r - 3, j - 1]), {j, 0, c - 1}]]], {c, 1, n + 1}], {r, 1, n + 1}]; SeriesCoefficient[((A[[n + 1]]. x^Range[0, n]) /. x -> x/(1 + x))/(1 + x), {x, 0, k}]];

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018, from PARI *)

(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).

Cf. A063007, A127674.

Sequence in context: A117427 A097761 A200756 * A122440 A046943 A107728

Adjacent sequences: A108553 A108554 A108555 * A108557 A108558 A108559

nonn,tabl

Paul D. Hanna, Jun 10 2005

approved