A128813 → basis A007318 → D+1

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Triangle begins:
  0:           1;
  1:           3,            1;
  2:          18,           16,             3;
  3:         174,          280,           135,            18;
  4:        2474,         6000,          5095,          1728,           180;
  5:       48824,       160980,        202585,        119193,         31500,          2700;
  6:     1281104,      5385192,       9118576,       7867584,       3574053,        777600,         56700;
  7:    43223312,    221841096,     475716960,     548422480,     362777373,     134871084,      25004700,      1587600;
  8:  1825635152,  11093826408,   28916113392,   42016610368,   36951982389,   19905010560,    6280626924,   1016064000,    57153600;
  9: 94448950112, 664153445328, 2043132574212, 3596474490508, 3972604195359, 2834624426985, 1291229437284, 354514739580, 50923857600, 2571912000;

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

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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.