A125662 → basis A013609 → θ+1

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Triangle begins:
  0:      1;
  1:      4,       4;
  2:     15,      32,      12;
  3:     56,     184,     144,      32;
  4:    209,     928,    1116,     512,      80;
  5:    780,    4364,    7104,    4992,    1600,     192;
  6:   2911,   19648,   40392,   38400,   18800,    4608,     448;
  7:  10864,   85872,  213408,  256320,  169600,   63360,   12544,   1024;
  8:  40545,  367296, 1070856, 1555968, 1300400,  655872,  197568,  32768,  2304;
  9: 151316, 1545492, 5171904, 8823168, 8921920, 5681088, 2308096, 581632, 82944, 5120;

The row polynomials satisfy -(6*n+2*n*x-n^3*x^2)*P(n,x) = -(12*n+12+(3*n^2+16*n+13)*x-(2*n^3+2*n^2-4*n-4)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (9+18*x-(3*n^2+3*n-11)*x^2-(4*n^2+4*n-2)*x^3-(n^2+n)*x^4)*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[(-(12*n + 12 + (3*n^2 + 16*n + 13)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (9 + 18*x - (3*n^2 + 3*n - 11)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-6*n - 2*n*x + n^3*x^2)]];
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] = (-(12*n + 12 + (3*n^2 + 16*n + 13)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (9 + 18*x - (3*n^2 + 3*n - 11)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-6*n - 2*n*x + n^3*x^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 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_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    • Geometric basis map (elementary) polytool ? step ›

      • from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of chebyshev_u?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from chebyshev_u?
            • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

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