A122542 → basis A013611 → θ+1

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Triangle begins:
   1:    1;
   2:    1,     8;
   3:    3,    32,     48;
   4:    7,   104,    336,     256;
   5:   17,   320,   1536,    2560,     1280;
   6:   41,   936,   5952,   15360,    16640,     6144;
   7:   99,  2656,  20976,   73216,   124160,    98304,    28672;
   8:  239,  7368,  69456,  305408,   707840,   878592,   544768,   131072;
   9:  577, 20096, 220032, 1166336,  3432960,  5849088,  5677056,  2883584,   589824;
  10: 1393, 54088, 674304, 4182016, 14932480, 32452608, 43294720, 34340864, 14745600, 2621440;

The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-4,x) - (4+24*x+32*x^2)*P(n-3,x) - (2+8*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 8*x, P(3,x) = 3 + 32*x + 48*x^2, P(4,x) = 7 + 104*x + 336*x^2 + 256*x^3.

P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[3, x_] := 3 + 32*x + 48*x^2;
P[4, x_] := 7 + 104*x + 336*x^2 + 256*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 4, x] - (4 + 24*x + 32*x^2)*P[n - 3, x] - (2 + 8*x + 16*x^2)*P[n - 2, x] + (4 + 8*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 + 8*x; v[3] = 3 + 32*x + 48*x^2; v[4] = 7 + 104*x + 336*x^2 + 256*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 8*x + 16*x^2)*v[i - 4] - (4 + 24*x + 32*x^2)*v[i - 3] - (2 + 8*x + 16*x^2)*v[i - 2] + (4 + 8*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 basis_map_A013611 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…?
    • 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 [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
    • 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.