A327997 → x ↦ x+1 → θ+1

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Triangle begins:
  0:       1;
  1:       4,        2;
  2:      17,       18,        3;
  3:      78,      130,       45,        4;
  4:     393,      904,      483,       88,        5;
  5:    2208,     6458,     4680,     1300,      150,       6;
  6:   13977,    48978,    44562,    16620,     2900,     234,      7;
  7:  100026,   401490,   434847,   202216,    47075,    5712,    343,     8;
  8:  806769,  3587904,  4449750,  2455152,   709695,  114408,  10290,   480,   9;
  9: 7280604, 35032626, 48342528, 30541640, 10455480, 2087442, 248724, 17328, 648, 10;

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) = 1, P(1,x) = 4 + 2*x, P(2,x) = 17 + 18*x + 3*x^2, P(3,x) = 78 + 130*x + 45*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 17 + 18*x + 3*x^2;
P[3, x_] := 78 + 130*x + 45*x^2 + 4*x^3;
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] = 1; v[2] = 4 + 2*x; v[3] = 17 + 18*x + 3*x^2; v[4] = 78 + 130*x + 45*x^2 + 4*x^3; 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023