A161198 → basis A007318 → (θ+1)²

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Triangle begins:
   1:         1;
   2:         3,          8;
   3:        15,         64,          36;
   4:       105,        568,         540,         128;
   5:       945,       5952,        7416,        3072,        400;
   6:     10395,      73032,      108360,       60160,      14000,       1152;
   7:    135135,    1032576,     1737324,     1167360,     370000,      55296,      3136;
   8:   2027025,   16569720,    30706452,    23687552,    9198000,    1895040,    197568,     8192;
   9:  34459425,  297901440,   596573424,   511866880,  230389600,   58705920,   8517376,   655360,   20736;
  10: 654729075, 5935802760, 12675451536, 11846620672, 5976986400, 1778934528, 321640704, 34701312, 2052864, 51200;

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

P[1, x_] := 1;
P[2, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (16*n - 24)*x - 36*x^2)*P[n - 2, x] + ((16*n - 24)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (4*n - 4 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 8*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (16*n - 24)*x - 36*x^2)*v[i - 2] + ((16*n - 24)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (4*n - 4 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  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 basis_map_A007318 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)
    • Affine, reversal and Möbius substitutions (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.