A122542 → D → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:    0;
   2:    1;
   3:    4,     2;
   4:   13,    14,     3;
   5:   40,    64,    30,     4;
   6:  117,   248,   180,    52,     5;
   7:  332,   874,   858,   388,    80,    6;
   8:  921,  2894,  3579,  2212,   715,  114,    7;
   9: 2512,  9168, 13668, 10728,  4760, 1188,  154,   8;
  10: 6761, 28096, 49008, 46664, 26410, 9060, 1834, 200, 9;

The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-4,x) - (4+6*x+2*x^2)*P(n-3,x) - (2+2*x+x^2)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 2*x, P(4,x) = 13 + 14*x + 3*x^2.

P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4 + 2*x;
P[4, x_] := 13 + 14*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 4, x] - (4 + 6*x + 2*x^2)*P[n - 3, x] - (2 + 2*x + x^2)*P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4 + 2*x; v[4] = 13 + 14*x + 3*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + 2*x + x^2)*v[i - 4] - (4 + 6*x + 2*x^2)*v[i - 3] - (2 + 2*x + x^2)*v[i - 2] + (4 + 2*x)*v[i - 1]); 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]
  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), Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Laguerre’s theorem on differential operators
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from derivative 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, 0,…?
    • Laguerre’s theorem on differential operators [SP14] 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 differential operators [SP14] 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)
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from shift_x_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,…?
    • Affine, reversal and Möbius substitutions (elementary) 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)
    • Affine, reversal and Möbius substitutions (elementary) 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)
    • Geometric basis map (elementary) 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)
    • Geometric basis map (elementary) 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. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.