A249250 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     7,      3;
  2:    22,     18,      3;
  3:    76,     96,     33,      3;
  4:   244,    420,    234,     48,      3;
  5:   772,   1692,   1320,    444,     63,     3;
  6:  2392,   6384,   6456,   3096,    726,    78,     3;
  7:  7312,  23040,  28728,  18072,   6084,  1080,    93,    3;
  8: 22096,  80352, 119376,  93456,  41580, 10620,  1506,  108,   3;
  9: 66160, 272880, 471168, 442656, 246636, 83556, 17040, 2004, 123, 3;

The row polynomials satisfy -(12*n-(3*n^3+12*n^2-21*n-24)*x-(12*n^2+2*n-36)*x^2-(n^2+10*n-13)*x^3-(n-1)*x^4)*P(n,x) = -(18*n^3+54*n^2+12*n+(15*n^3+105*n^2+102*n+24)*x+(60*n^2+148*n+36)*x^2+(5*n^2+62*n+13)*x^3+(5*n+1)*x^4)*P(n-1,x) + (36*n^2+180*n+216+(72*n^2+504*n+780)*x+(39*n^2+495*n+1086)*x^2+(3*n^2+195*n+719)*x^3+(25*n+222)*x^4+(n+26)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*P[n - 1, x] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*D[P[n - 1, x], x])/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(18*n^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*v[i - 1] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*deriv(v[i - 1], x))/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*x^4)); 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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 with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (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 with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (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 with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (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 with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (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 with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.