A116412 → basis A013610 → basis A007318

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Triangle begins:
  0:        1;
  1:        7,         3;
  2:       46,        42,         9;
  3:      304,       423,       189,        27;
  4:     2008,      3756,      2565,       756,        81;
  5:    13264,     31164,     28683,     12906,      2835,      243;
  6:    87616,    247824,    287118,    174204,     58320,    10206,      729;
  7:   578752,   1914240,   2674404,   2044251,    922590,   245673,    35721,    2187;
  8:  3822976,  14475840,  23681088,  21846888,  12424185,  4457592,   984879,  122472,   6561;
  9: 25252864, 107717184, 201954384, 218324808, 149908887, 67768569, 20161953, 3805380, 413343, 19683;

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-2,x) + (6+3*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[(4 + 3*x)*P[n - 2, x] + (6 + 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] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 3*x)*v[i - 2] + (6 + 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 basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    • Geometric basis map (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 [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 (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) 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) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (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) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

• A116412 → basis A013610 → 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