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 A108556 Triangle, read by rows, where row n equals the inverse binomial transform of the crystal ball sequence for D_n lattice. 5
 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, 1, 144, 4050, 42588, 221544, 643824, 1096032, 1085760, 579456, 128768 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS 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. FORMULA 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 EXAMPLE 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; ... MATHEMATICA 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 *) PROG (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) CROSSREFS 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 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jun 10 2005 STATUS approved

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Last modified April 21 23:52 EDT 2021. Contains 343156 sequences. (Running on oeis4.)