A122848 → basis A013609 → D

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Triangle begins:
  0:     0;
  1:     2;
  2:     6,      8;
  3:    18,     48,      24;
  4:    56,    216,     240,      64;
  5:   180,    920,    1560,     960,     160;
  6:   612,   3840,    8760,    8640,    3360,     384;
  7:  2156,  16128,   46200,   62720,   40320,   10752,    896;
  8:  7968,  68768,  236544,  409920,  371840,  166656,  32256,  2048;
  9: 30456, 298656, 1199520, 2528064, 2973600, 1919232, 628992, 92160, 4608;

The row polynomials satisfy (n^2-3*n+2-(4*n-4)*x)*P(n,x) = -(n^3-7*n^2+10*n-(6*n^2-18*n)*x+8*n*x^2)*P(n-1,x) + (n^2-3*n+(2*n^2-10*n)*x-8*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 10*n - (6*n^2 - 18*n)*x + 8*n*x^2)*P[n - 1, x] + (n^2 - 3*n + (2*n^2 - 10*n)*x - 8*n*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (4*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 7*n^2 + 10*n - (6*n^2 - 18*n)*x + 8*n*x^2)*v[i - 1] + (n^2 - 3*n + (2*n^2 - 10*n)*x - 8*n*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (4*n - 4)*x)); 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_A013609 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.