A086645 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
   1:         1;
   2:         6,          8;
   3:        56,        128,          64;
   4:       576,       1920,        1920,         512;
   5:      6016,      26624,       40960,       24576,         4096;
   6:     62976,     348160,      727040,      696320,       286720,        32768;
   7:    659456,    4374528,    11550720,    15269888,     10321920,      3145728,      262144;
   8:   6905856,   53444608,   170655744,   288096256,    272498688,    139460608,    33030144,     2097152;
   9:  72318976,  639631360,  2395996160,  4924112896,   5997854720,   4345298944,  1761607680,   335544320,   16777216;
  10: 757334016, 7535591424, 32393134080, 78473330688, 117039955968, 109999816704, 63770198016, 21139292160, 3321888768, 134217728;

The row polynomials satisfy P(n,x) = -(16+64*x+64*x^2)*P(n-2,x) + (12+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 8*x.

P[1, x_] := 1;
P[2, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 64*x + 64*x^2)*P[n - 2, x] + (12 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 8*x; for(n = 3, N, my(i = n); v[i] = -(16 + 64*x + 64*x^2)*v[i - 2] + (12 + 16*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, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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 Triangle read by rows: T(n, k) = binomial(2n, 2k).?
            • ↑ Three-term recurrence criterion (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A084534 → basis A013609

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023