A122542 → θ+1 → x ↦ x+1

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Triangle begins:
   1:    1;
   2:    2,     2;
   3:    7,    10,     3;
   4:   20,    40,    24,      4;
   5:   57,   144,   126,     44,     5;
   6:  158,   482,   552,    292,    70,     6;
   7:  431,  1538,  2169,   1532,   565,   102,     7;
   8: 1160,  4736,  7920,   6984,  3480,   972,   140,    8;
   9: 3089, 14192, 27420,  28952, 18170,  6900,  1540,  184,   9;
  10: 8154, 41618, 91152, 112008, 84740, 40752, 12404, 2296, 234, 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) = 2 + 2*x, P(3,x) = 7 + 10*x + 3*x^2, P(4,x) = 20 + 40*x + 24*x^2 + 4*x^3.

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 7 + 10*x + 3*x^2;
P[4, x_] := 20 + 40*x + 24*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] = 2 + 2*x; v[3] = 7 + 10*x + 3*x^2; v[4] = 20 + 40*x + 24*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 (−∞, -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)
• 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.