A132393 → θ+1 → x ↦ x/(x+1)

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Triangle begins:
  0: 1;
  1: 0,     2;
  2: 0,     2,      5;
  3: 0,     4,     17,      17;
  4: 0,    12,     69,     126,       74;
  5: 0,    48,    342,     878,      972,      394;
  6: 0,   240,   2022,    6588,    10457,     8128,     2484;
  7: 0,  1440,  13932,   54556,   111379,   125571,    74356,    18108;
  8: 0, 10080, 109764,  499432,  1237345,  1809300,  1565624,   743816,  149904;
  9: 0, 80640, 973872, 5031680, 14591028, 26055354, 29406808, 20521140, 8106552, 1389456;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n^2-4*n+4+(4*n^2-14*n+12)*x+(6*n^2-18*n+13)*x^2+(4*n^2-10*n+6)*x^3+(n^2-2*n+1)*x^4)*P(n-2,x) + (2*n-3+(6*n-7)*x+(6*n-5)*x^2+(2*n-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 4 + (4*n^2 - 14*n + 12)*x + (6*n^2 - 18*n + 13)*x^2 + (4*n^2 - 10*n + 6)*x^3 + (n^2 - 2*n + 1)*x^4)*P[n - 2, x] + (2*n - 3 + (6*n - 7)*x + (6*n - 5)*x^2 + (2*n - 1)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 4 + (4*n^2 - 14*n + 12)*x + (6*n^2 - 18*n + 13)*x^2 + (4*n^2 - 10*n + 6)*x^3 + (n^2 - 2*n + 1)*x^4)*v[i - 2] + (2*n - 3 + (6*n - 7)*x + (6*n - 5)*x^2 + (2*n - 1)*x^3)*v[i - 1])/(1 + 2*x + 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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) 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.