A091044 → D → θ+1

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Triangle begins:
  0:   0;
  1:   2;
  2:  10,    12;
  3:  28,   112,     36;
  4:  60,   504,    540,      80;
  5: 110,  1584,   3564,    1760,     150;
  6: 182,  4004,  15444,   16016,    4550,     252;
  7: 280,  8736,  51480,   91520,   54600,   10080,    392;
  8: 408, 17136, 143208,  388960,  397800,  154224,  19992,   576;
  9: 570, 31008, 348840, 1343680, 2099500, 1395360, 379848, 36480, 810;

The row polynomials satisfy (2*n^2-3*n+1+(4*n^4-12*n^3+13*n^2-6*n+1)*x)*P(n,x) = (2*n^2+3*n+1+(4*n^4+7*n^2-14*n-1)*x+(12*n^4-16*n^3-5*n^2+3*n+4)*x^2)*P(n-1,x) + (4*n*x+(8*n^3-6*n-2)*x^2-(8*n^3-2*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*P[n - 1, x] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*v[i - 1] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x)); 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from derivative of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
        • 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 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
            • ↑ Three-term recurrence criterion (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
        • ↑ 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) (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