A161198 → basis A008277 → (θ+1)²

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Triangle begins:
   1:         1;
   2:         3,           8;
   3:        15,          80,          36;
   4:       105,         840,         756,         128;
   5:       945,       10080,       13608,        4608,         400;
   6:     10395,      138600,      249480,      126720,       22000,       1152;
   7:    135135,     2162160,     4864860,     3294720,      858000,      89856,       3136;
   8:   2027025,    37837800,   102162060,    86486400,    30030000,    4717440,     329280,     8192;
   9:  34459425,   735134400,  2315673360,  2352430080,  1021020000,  213857280,   22391040,  1114112,   20736;
  10: 654729075, 15713497800, 56568592080, 67044257280, 34918884000, 9142398720, 1276289280, 95256576, 3545856, 51200;

The row polynomials satisfy (1-(10*n-12)*x+(4*n^2-8*n+4)*x^2)*P(n,x) = (2*n-1-(20*n^2-34*n+4)*x+(8*n^3-20*n^2-20*n+4)*x^2+8*n^2*x^3)*P(n-1,x) + (2*x-(20*n-4)*x^2+8*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 - (20*n^2 - 34*n + 4)*x + (8*n^3 - 20*n^2 - 20*n + 4)*x^2 + 8*n^2*x^3)*P[n - 1, x] + (2*x - (20*n - 4)*x^2 + 8*n^2*x^3)*D[P[n - 1, x], x])/(1 - (10*n - 12)*x + (4*n^2 - 8*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 - (20*n^2 - 34*n + 4)*x + (8*n^3 - 20*n^2 - 20*n + 4)*x^2 + 8*n^2*x^3)*v[i - 1] + (2*x - (20*n - 4)*x^2 + 8*n^2*x^3)*deriv(v[i - 1], x))/(1 - (10*n - 12)*x + (4*n^2 - 8*n + 4)*x^2)); 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_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.