A122542 → θ+1 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
   1: 1;
   2: 0, 2;
   3: 0, 4,   7;
   4: 0, 4,  20,   20;
   5: 0, 4,  36,   84,   57;
   6: 0, 4,  52,  204,  308,   158;
   7: 0, 4,  68,  384,  944,  1048,   431;
   8: 0, 4,  84,  624, 2176,  3864,  3384,  1160;
   9: 0, 4, 100,  924, 4220, 10520, 14568, 10520,  3089;
  10: 0, 4, 116, 1284, 7292, 23680, 45688, 51752, 31768, 8154;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(x^2+4*x^3+6*x^4+4*x^5+x^6)*P(n-4,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*P(n-3,x) - (1+4*x+7*x^2+6*x^3+2*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 4*x + 7*x^2, P(4,x) = 4*x + 20*x^2 + 20*x^3.

P[1, x_] := 1;
P[2, x_] := 2*x;
P[3, x_] := 4*x + 7*x^2;
P[4, x_] := 4*x + 20*x^2 + 20*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*P[n - 4, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*P[n - 3, x] - (1 + 4*x + 7*x^2 + 6*x^3 + 2*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 4*x + 7*x^2; v[4] = 4*x + 20*x^2 + 20*x^3; for(n = 5, N, my(i = n); v[i] = (-(x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*v[i - 4] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*v[i - 3] - (1 + 4*x + 7*x^2 + 6*x^3 + 2*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + 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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (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.