A034807 → basis A013610

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Submission text ?

Triangle begins:
  0:  2;
  1:  1;
  2:  3,   6;
  3:  4,   9;
  4:  7,  24,   18;
  5: 11,  45,   45;
  6: 18,  90,  135,   54;
  7: 29, 168,  315,  189;
  8: 47, 312,  720,  648, 162;
  9: 76, 567, 1539, 1782, 729;

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.

P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 2, x] + P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 2] + v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003