A122848 → basis A013610 → D+1

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Triangle begins:
  0:     1;
  1:     4,      3;
  2:    11,     27,       9;
  3:    31,    135,     135,       27;
  4:    94,    570,    1053,      594,       81;
  5:   296,   2340,    6300,     6615,     2430,      243;
  6:   994,   9558,   33885,    53595,    36450,     9477,      729;
  7:  3466,  39522,  174069,   369495,   385560,   183708,    35721,    2187;
  8: 12716, 166680,  875700,  2341332,  3342465,  2463048,   867510,  131220,   6561;
  9: 48304, 717660, 4384368, 14147784, 25780356, 26377407, 14390460, 3897234, 472392, 19683;

The row polynomials satisfy (27*n^2-45*n-6-(54*n+36)*x-54*x^2)*P(n,x) = -(27*n^3-153*n^2+138*n+84-(135*n^2-207*n-360)*x+(108*n+378)*x^2+162*x^3)*P(n-1,x) + (18*n^2-30*n-16+(54*n^2-126*n-108)*x-(108*n+216)*x^2-108*x^3)*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[(-(27*n^3 - 153*n^2 + 138*n + 84 - (135*n^2 - 207*n - 360)*x + (108*n + 378)*x^2 + 162*x^3)*P[n - 1, x] + (18*n^2 - 30*n - 16 + (54*n^2 - 126*n - 108)*x - (108*n + 216)*x^2 - 108*x^3)*D[P[n - 1, x], x])/(27*n^2 - 45*n - 6 - (54*n + 36)*x - 54*x^2)]];
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] = (-(27*n^3 - 153*n^2 + 138*n + 84 - (135*n^2 - 207*n - 360)*x + (108*n + 378)*x^2 + 162*x^3)*v[i - 1] + (18*n^2 - 30*n - 16 + (54*n^2 - 126*n - 108)*x - (108*n + 216)*x^2 - 108*x^3)*deriv(v[i - 1], x))/(27*n^2 - 45*n - 6 - (54*n + 36)*x - 54*x^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 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_A013610 of Exponential Riordan array (1, x(1+x/2)).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).?
            • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

              • from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
                • 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 read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
                    • ↑ 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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
                        • ↑ 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 Exponential Riordan array (1, x(1+x/2)).?
                • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Exponential Riordan array (1, x(1+x/2)).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Parent-row provers of polytool polytool ? 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
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Exponential Riordan array (1, x(1+x/2)).?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (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.