A137320 → reverse → (θ+1)²

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Triangle begins:
  0:   1;
  1:   2;
  2:   4,     8;
  3:   8,    48,     36;
  4:  16,   192,    396,     192;
  5:  32,   640,   2520,    3200,     1200;
  6:  64,  1920,  12240,   28800,    27400,     8640;
  7: 128,  5376,  50400,  188160,   324800,   254016,    70560;
  8: 256, 14336, 185472, 1003520,  2707600,  3782016,  2561328,   645120;
  9: 512, 36864, 628992, 4644864, 17959200, 38755584, 46304608, 28053504, 6531840;

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-4+(8*n^2-24*n+16)*x-(9*n^3-45*n^2+72*n-36)*x^2)*P(n-2,x) + ((8*n^2-24*n+16)*x^2+(4*n^3-20*n^2+32*n-16)*x^3)*d/dx P(n-2,x) + (4*n-6+(4*n^2-12*n+8)*x)*P(n-1,x) - (4*n^2-12*n+8)*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4 + 8*x.

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (8*n^2 - 24*n + 16)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*P[n - 2, x] + ((8*n^2 - 24*n + 16)*x^2 + (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*D[P[n - 2, x], x] + (4*n - 6 + (4*n^2 - 12*n + 8)*x)*P[n - 1, x] - (4*n^2 - 12*n + 8)*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] = 2; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 4 + (8*n^2 - 24*n + 16)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*v[i - 2] + ((8*n^2 - 24*n + 16)*x^2 + (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*deriv(v[i - 2], x) + (4*n - 6 + (4*n^2 - 12*n + 8)*x)*v[i - 1] - (4*n^2 - 12*n + 8)*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 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.