A161198 → basis A013611 → D

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Triangle begins:
   1:          0;
   2:          8;
   3:         64,         128;
   4:        568,        1920,         1536;
   5:       5952,       26368,        36864,        16384;
   6:      73032,      385280,       721920,       573440,       163840;
   7:    1032576,     6177152,     14008320,     15155200,      7864320,      1572864;
   8:   16569720,   109178496,    284250624,    376750080,    269516800,     99090432,    14680064;
   9:  297901440,  2121149952,   6142402560,   9436758016,   8349286400,   4271898624,  1174405120,   134217728;
  10: 5935802760, 45068272128, 142159448064, 244817362944, 253004021760, 161319223296, 62184751104, 13287555072, 1207959552;

The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(32*n-48)*x+64*x^2)*P(n-2,x) + (4*n-4+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 8.

P[1, x_] := 0;
P[2, x_] := 8;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*P[n - 2, x] + (4*n - 4 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 8; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*v[i - 2] + (4*n - 4 + 16*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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.