A327997 → x ↦ x+1 → D

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Triangle begins:
  0:        0;
  1:        1;
  2:        9,        2;
  3:       65,       30,        3;
  4:      452,      322,       66,       4;
  5:     3229,     3120,      975,     120,       5;
  6:    24489,    29708,    12465,    2320,     195,      6;
  7:   200745,   289898,   151662,   37660,    4760,    294,     7;
  8:  1793952,  2966500,  1841364,  567756,   95340,   8820,   420,   8;
  9: 17516313, 32228352, 22906230, 8364384, 1739535, 213192, 15162, 576, 9;

The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-3*n-18+(6*n-12)*x)*P(n-3,x) - (n^2+10*n-5+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9 + 2*x, P(3,x) = 65 + 30*x + 3*x^2.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 9 + 2*x;
P[3, x_] := 65 + 30*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 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] = 0; v[2] = 1; v[3] = 9 + 2*x; v[4] = 65 + 30*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*v[i - 3] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
        • ↑ 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: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023