A249251 → basis A007318 → basis A007318

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Submission text ?

Triangle begins:
  0:       1;
  1:       7,        3;
  2:      37,       30,        6;
  3:     199,      237,       93,       12;
  4:    1069,     1680,      981,      252,       24;
  5:    5743,    11211,     8688,     3339,      636,      48;
  6:   30853,    71970,    69504,    35556,    10158,    1536,      96;
  7:  165751,   449721,   520047,   332154,   126513,   28728,    3600,    192;
  8:  890461,  2754900,  3710655,  2841480,  1352745,  409896,   77184,   8256,   384;
  9: 4783807, 16620615, 25552890, 22813065, 13031865, 4938939, 1241640, 199632, 18624, 768;

The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + (5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.

P[0, x_] := 1;
P[1, x_] := 7 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + (5 + 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] = 1; v[2] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + x)*v[i - 2] + (5 + 2*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 basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Geometric basis map (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 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
            • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
            • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
            • 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) polytool ? step ›

                  • from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
                    • ↑ Liu–Wang theorem for three-term recurrences (above)
                    • ↑ Liu–Wang theorem with a lag row (above)
                    • 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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • 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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • 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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

• A249251 → basis A007318 → x ↦ x+1 15 rows; no all-index equality certificate was found

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