A035607 → reverse → x ↦ x+1

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Submission text ?

Triangle begins:
   1:    1;
   2:    3,     1;
   3:    7,     6,     1;
   4:   17,    23,     9,     1;
   5:   41,    76,    48,    12,     1;
   6:   99,   233,   204,    82,    15,    1;
   7:  239,   682,   765,   428,   125,   18,    1;
   8:  577,  1935,  2649,  1907,   775,  177,   21,   1;
   9: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;
  10: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;

The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-9*n+10+(2*n^2-12*n+16)*x-(2*n-4)*x^2)*P(n-1,x) - (8*n-28+(16*n-59)*x+(9*n-36)*x^2+(n-5)*x^3)*d/dx P(n-1,x) + (8+24*x+25*x^2+10*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 16)*x - (2*n - 4)*x^2)*P[n - 1, x] - (8*n - 28 + (16*n - 59)*x + (9*n - 36)*x^2 + (n - 5)*x^3)*D[P[n - 1, x], x] + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 16)*x - (2*n - 4)*x^2)*v[i - 1] - (8*n - 28 + (16*n - 59)*x + (9*n - 36)*x^2 + (n - 5)*x^3)*deriv(v[i - 1], x) + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n - 2)); 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 reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 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 reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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