A075497 → reverse → (θ+1)²

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Triangle begins:
  0: 1;
  1: 1,   8;
  2: 1,  24,    36;
  3: 1,  48,   252,    128;
  4: 1,  80,   900,   1920,     400;
  5: 1, 120,  2340,  11520,   12400,     1152;
  6: 1, 168,  5040,  44800,  120400,    72576,      3136;
  7: 1, 224,  9576, 134400,  680400,  1112832,    398272,     8192;
  8: 1, 288, 16632, 338688, 2780400,  8951040,   9486400,  2088960,    20736;
  9: 1, 360, 27000, 752640, 9130800, 48988800, 106953280, 76431360, 10596096, 51200;

The row polynomials satisfy n^2*P(n,x) = -(n^2+n+(8*n^3-8)*x-(36*n^4-8*n^3-28*n^2)*x^2)*P(n-2,x) - (x-(8*n^3+8*n^2+10)*x^2-(16*n^4-56*n^3-52*n^2)*x^3)*d/dx P(n-2,x) + (2*x^3-(16*n^3+12*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+n+8*n^3*x)*P(n-1,x) + (x-(8*n^3+8*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (8*n^3 - 8)*x - (36*n^4 - 8*n^3 - 28*n^2)*x^2)*P[n - 2, x] - (x - (8*n^3 + 8*n^2 + 10)*x^2 - (16*n^4 - 56*n^3 - 52*n^2)*x^3)*D[P[n - 2, x], x] + (2*x^3 - (16*n^3 + 12*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + n + 8*n^3*x)*P[n - 1, x] + (x - (8*n^3 + 8*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (8*n^3 - 8)*x - (36*n^4 - 8*n^3 - 28*n^2)*x^2)*v[i - 2] - (x - (8*n^3 + 8*n^2 + 10)*x^2 - (16*n^4 - 56*n^3 - 52*n^2)*x^3)*deriv(v[i - 2], x) + (2*x^3 - (16*n^3 + 12*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + n + 8*n^3*x)*v[i - 1] + (x - (8*n^3 + 8*n^2)*x^2)*deriv(v[i - 1], x))/(n^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 reversal of Stirling2 triangle with scaled diagonals (powers of 2).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Stirling2 triangle with scaled diagonals (powers of 2).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Stirling2 triangle with scaled diagonals (powers of 2).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Stirling2 triangle with scaled diagonals (powers of 2).?
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (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. [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.