A249252 → basis A007318 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    1;
  1:    4,     5;
  2:    8,    24,     17;
  3:   16,    80,    124,     61;
  4:   32,   224,    552,    576,     217;
  5:   64,   576,   1968,   3216,    2532,     773;
  6:  128,  1408,   6176,  13888,   16968,   10728,    2753;
  7:  256,  3328,  17856,  51392,   85968,   83856,   44300,    9805;
  8:  512,  7680,  48768, 171520,  366240,  487328,  395560,  179472,  34921;
  9: 1024, 17408, 127744, 531712, 1385536, 2348096, 2592752, 1801936, 716468, 124373;

The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(8*n^2-12*n+4)*x+(20*n^2-64*n+44)*x^2+(17*n^2-51*n+34)*x^3)*P(n-1,x) - ((4*n-4)*x+(28*n-54)*x^2+(57*n-118)*x^3+(34*n-68)*x^4)*d/dx P(n-1,x) + (4*x^2+24*x^3+37*x^4+17*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x.

P[0, x_] := 1;
P[1, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n^2 - 12*n + 4)*x + (20*n^2 - 64*n + 44)*x^2 + (17*n^2 - 51*n + 34)*x^3)*P[n - 1, x] - ((4*n - 4)*x + (28*n - 54)*x^2 + (57*n - 118)*x^3 + (34*n - 68)*x^4)*D[P[n - 1, x], x] + (4*x^2 + 24*x^3 + 37*x^4 + 17*x^5)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n^2 - 12*n + 4)*x + (20*n^2 - 64*n + 44)*x^2 + (17*n^2 - 51*n + 34)*x^3)*v[i - 1] - ((4*n - 4)*x + (28*n - 54)*x^2 + (57*n - 118)*x^3 + (34*n - 68)*x^4)*deriv(v[i - 1], x) + (4*x^2 + 24*x^3 + 37*x^4 + 17*x^5)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

• A249252 → reverse → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A249252 → basis A007318
• A035607 → basis A007318 → basis A013609

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