A249248 → reverse → basis A013610

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Submission text ?

Triangle begins:
  0:      1;
  1:      4,       3;
  2:     11,      27,       9;
  3:     38,     147,     135,       27;
  4:    131,     708,    1161,      594,       81;
  5:    490,    3375,    7902,     7263,     2430,      243;
  6:   1897,   15927,   48924,    66771,    39690,     9477,      729;
  7:   7714,   76305,  289845,   528066,   473445,   198288,    35721,    2187;
  8:  32501,  370314, 1678950,  3847554,  4691115,  2973834,   928746,  131220,   6561;
  9: 142142, 1832601, 9656442, 26707968, 41501322, 36207243, 17084844, 4142178, 472392, 19683;

The row polynomials satisfy P(n,x) = (n+1+(3*n+3)*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (2 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (2 + 3*x)*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 reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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