A103327 → basis A013609 → reverse

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Triangle begins:
  0:   1;
  1:   2,     4;
  2:   4,    24,     16;
  3:   8,    96,    160,      64;
  4:  16,   320,    960,     896,      256;
  5:  32,   960,   4480,    7168,     4608,     1024;
  6:  64,  2688,  17920,   43008,    46080,    22528,     4096;
  7: 128,  7168,  64512,  215040,   337920,   270336,   106496,   16384;
  8: 256, 18432, 215040,  946176,  2027520,  2342912,  1490944,  491520,   65536;
  9: 512, 46080, 675840, 3784704, 10543104, 16400384, 14909440, 7864320, 2228224, 262144;

The row polynomials satisfy (2*n-1)*P(n,x) = (4*n-2+4*n*x)*P(n-1,x) + (8*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 2 + 4*n*x)*P[n - 1, x] + (8*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n - 2 + 4*n*x)*v[i - 1] + (8*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A103327 → basis A013609
• Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
• A123519 → reverse
• A086645 → basis A013609 → D

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