A116412 → basis A007318 → basis A013609

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Triangle begins:
  0:      1;
  1:      5,       2;
  2:     22,      20,       4;
  3:     98,     138,      60,       8;
  4:    436,     832,     564,     160,      16;
  5:   1940,    4672,    4316,    1904,     400,      32;
  6:   8632,   25104,   29400,   17696,    5760,     960,     64;
  7:  38408,  130904,  185784,  142024,   63040,   16224,   2240,    128;
  8: 170896,  667904, 1113952, 1033856,  583120,  204416,  43456,   5120,   256;
  9: 760400, 3352032, 6424992, 7018944, 4810320, 2142432, 619584, 112128, 11520, 512;

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.

P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (4 + 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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (4 + 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 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)
    • 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)
    • 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)

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