A082985 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,   4;
  2:  5,  15,   11;
  3:  7,  35,   56,    29;
  4:  9,  66,  171,   189,    76;
  5: 11, 110,  407,   715,   605,    199;
  6: 13, 169,  832,  2054,  2730,   1872,    521;
  7: 15, 245, 1533,  4965,  9180,   9810,   5655,   1364;
  8: 17, 340, 2618, 10642, 25585,  37774,  33745,  16779,  3571;
  9: 19, 456, 4218, 20862, 62453, 119168, 146357, 112309, 49096, 9349;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1-(n-5)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*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[((2*n + 1 - (n - 5)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*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] = ((2*n + 1 - (n - 5)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*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_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
        • 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 reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from shift_x_plus_one of Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q…?
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…?
                    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
        • ↑ 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) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
        • ↑ 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) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

• A082985 → reverse → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A082985 → x ↦ x+1
• A111125 → x ↦ x/(x+1)
• A111125 → reverse → x ↦ x+1

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