A122542 → x ↦ x+1 → θ+1

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Triangle begins:
   1:    1;
   2:    1,     2;
   3:    3,     8,     3;
   4:    7,    26,    21,     4;
   5:   17,    80,    96,    40,     5;
   6:   41,   234,   372,   240,    65,     6;
   7:   99,   664,  1311,  1144,   485,    96,     7;
   8:  239,  1842,  4341,  4772,  2765,   858,   133,    8;
   9:  577,  5024, 13752, 18224, 13410,  5712,  1386,  176,   9;
  10: 1393, 13522, 42144, 65344, 58330, 31692, 10570, 2096, 225, 10;

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) = 1, P(2,x) = 1 + 2*x, P(3,x) = 3 + 8*x + 3*x^2, P(4,x) = 7 + 26*x + 21*x^2 + 4*x^3.

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 3 + 8*x + 3*x^2;
P[4, x_] := 7 + 26*x + 21*x^2 + 4*x^3;
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] = 1; v[2] = 1 + 2*x; v[3] = 3 + 8*x + 3*x^2; v[4] = 7 + 26*x + 21*x^2 + 4*x^3; 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 (−∞, 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 [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
    • 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. [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.