A132393 → θ+1 → D

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Triangle begins:
  0:     0;
  1:     2;
  2:     2,      6;
  3:     4,     18,      12;
  4:    12,     66,      72,      20;
  5:    48,    300,     420,     200,     30;
  6:   240,   1644,    2700,    1700,    450,     42;
  7:  1440,  10584,   19488,   14700,   5250,    882,    56;
  8: 10080,  78408,  157584,  135380,  58800,  13524,  1568,   72;
  9: 80640, 657504, 1417488, 1345680, 673470, 190512, 30576, 2592, 90;

The row polynomials satisfy P(n,x) = (n^3-9*n^2+27*n-27+(3*n^2-18*n+27)*x+(3*n-9)*x^2+x^3)*P(n-3,x) - (3*n^2-15*n+19+(6*n-15)*x+3*x^2)*P(n-2,x) + (3*n-6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2 + 6*x.

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(n^3 - 9*n^2 + 27*n - 27 + (3*n^2 - 18*n + 27)*x + (3*n - 9)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 15*n + 19 + (6*n - 15)*x + 3*x^2)*P[n - 2, x] + (3*n - 6 + 3*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] = 2; v[3] = 2 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 9*n^2 + 27*n - 27 + (3*n^2 - 18*n + 27)*x + (3*n - 9)*x^2 + x^3)*v[i - 3] - (3*n^2 - 15*n + 19 + (6*n - 15)*x + 3*x^2)*v[i - 2] + (3*n - 6 + 3*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

Proof chains ? JSON

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

  • from theta_plus_one of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.