A249248 → reverse → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 3,   2;
  2: 3,  14,    3;
  3: 3,  44,   36,     4;
  4: 3,  80,  207,    72,      5;
  5: 3, 122,  723,   636,    125,      6;
  6: 3, 170, 1746,  3532,   1550,    198,     7;
  7: 3, 224, 3465, 13572,  12325,   3258,   294,     8;
  8: 3, 284, 6096, 39144,  69025,  34788,  6174,   416,   9;
  9: 3, 350, 9882, 93472, 287605, 265518, 84770, 10832, 567, 10;

The row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 3 + 14*x + 3*x^2, P(3,x) = 3 + 44*x + 36*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 3 + 14*x + 3*x^2;
P[3, x_] := 3 + 44*x + 36*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 3 + 14*x + 3*x^2; v[4] = 3 + 44*x + 36*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 + n^2 - 2*n)*x^2*v[i - 4] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*v[i - 3] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*v[i - 2] + (2*n + 3 + (2*n + 2)*x)*v[i - 1])/(n + 1)); 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.