A122848 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     3,      2;
  2:     8,     14,       4;
  3:    22,     66,      48,       8;
  4:    66,    272,     348,     144,      16;
  5:   206,   1100,    2020,    1480,     400,      32;
  6:   688,   4452,   10680,   11560,    5520,    1056,     64;
  7:  2388,  18284,   54264,   78120,   56000,   18816,   2688,    128;
  8:  8732,  76736,  270928,  488768,  474320,  241024,  60032,   6656,   256;
  9: 33076, 329112, 1348848, 2927904, 3605616, 2513952, 948864, 182016, 16128, 512;

The row polynomials satisfy P(n,x) = -(n-4-2*x)*P(n-1,x) + (1+2*x)*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[-(n - 4 - 2*x)*P[n - 1, x] + (1 + 2*x)*D[P[n - 1, x], x]];
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] = -(n - 4 - 2*x)*v[i - 1] + (1 + 2*x)*deriv(v[i - 1], 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.