A358735 → basis A007318 → D+1

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Triangle begins:
  0:        1;
  1:        3,         1;
  2:       15,        12,          2;
  3:       93,       128,         52,          6;
  4:      675,      1368,        958,        268,         24;
  5:     5587,     15243,      15704,       7520,       1628,       120;
  6:    51845,    179796,     249012,     174564,      64288,     11448,       720;
  7:   532705,   2257064,    3967648,    3726444,    1999448,    602824,     91656,     5040;
  8:  6002125,  30187204,   64744876,   76991156,   55154608,  24128920,   6189440,   823968,   40320;
  9: 73576351, 429754749, 1092786520, 1582489552, 1431699396, 833669512, 309268720, 69276240, 8222112, 362880;

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (n^4-n^2-6*n+2+(2*n^4-4*n^3-2*n^2+4*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (2*n^3-2*n^2-6*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 15 + 12*x + 2*x^2, P(3,x) = 93 + 128*x + 52*x^2 + 6*x^3.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 15 + 12*x + 2*x^2;
P[3, x_] := 93 + 128*x + 52*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 15 + 12*x + 2*x^2; v[4] = 93 + 128*x + 52*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
        • ↑ Three-term recurrence criterion (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