A122542 → reverse → x ↦ x+1

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Submission text ?

Triangle begins:
   1:    1;
   2:    1;
   3:    3,    2;
   4:    7,    8,     2;
   5:   17,   28,    14,    2;
   6:   41,   88,    66,   20,    2;
   7:   99,  262,   262,  122,   26,    2;
   8:  239,  752,   940,  592,  196,   32,   2;
   9:  577, 2104,  3156, 2508, 1132,  288,  38,  2;
  10: 1393, 5776, 10108, 9704, 5560, 1936, 398, 44, 2;

The row polynomials satisfy (n-3)*P(n,x) = (2*n^2-9*n+9)*P(n-1,x) - (8*n-28+(8*n-31)*x+(n-5)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 3 + 2*x.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 9*n + 9)*P[n - 1, x] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 3 + 2*x; for(n = 4, N, my(i = n); v[i] = ((2*n^2 - 9*n + 9)*v[i - 1] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 3)); 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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
  by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

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

  • from reversal 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, 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)

Related ?

Candidate equalities ?

• A122542 → reverse → basis A007318 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A035607 → basis A007318
• A113413 → x ↦ x/(x+1)
• A122542 → x ↦ x/(x+1)
• A035607 → reverse → x ↦ x/(x+1)
• A113413 → reverse → x ↦ x+1
• A035607 → x ↦ x+1

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