A137320 → reverse → D+1

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Triangle begins:
  0:    1;
  1:    2;
  2:    6,      2;
  3:   20,     20,      4;
  4:   64,    136,     80,      12;
  5:  192,    720,    880,     392,      48;
  6:  544,   3200,   6760,    6184,    2296,     240;
  7: 1472,  12544,  40880,   63728,   48272,   15696,    1440;
  8: 3840,  44800, 208768,  495936,  633584,  418688,  122832,   10080;
  9: 9728, 148992, 940800, 3163776, 6101088, 6746496, 4013344, 1083456, 80640;

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

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (4*n^2 - 12*n + 8)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*P[n - 2, x] + (4*n - 6 + (2*n^2 - 6*n + 4)*x)*P[n - 1, 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] = 2; v[3] = 6 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 4 + (4*n^2 - 12*n + 8)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*v[i - 2] + (4*n - 6 + (2*n^2 - 6*n + 4)*x)*v[i - 1])/(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 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 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n …?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • ↑ 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.