A075497 → reverse → D

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Triangle begins:
  0:  0;
  1:  2;
  2:  6,    8;
  3: 12,   56,     24;
  4: 20,  200,    360,      64;
  5: 30,  520,   2160,    1984,     160;
  6: 42, 1120,   8400,   19264,   10080,      384;
  7: 56, 2128,  25200,  108864,  154560,    48768,     896;
  8: 72, 3696,  63504,  444864, 1243200,  1161600,  228480,    2048;
  9: 90, 6000, 141120, 1460928, 6804000, 13096320, 8359680, 1046528, 4608;

The row polynomials satisfy (n-1)*P(n,x) = -(n+(4*n^2-8*n)*x+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (2*n*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) + (2*n-1+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n-2)*x^2*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[(-(n + (4*n^2 - 8*n)*x + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (2*n*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n - 2)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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] = (-(n + (4*n^2 - 8*n)*x + (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (2*n*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n - 2)*x^2*deriv(v[i - 1], x))/(n - 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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.