A122848 → basis A013609 → reverse

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Triangle begins:
  0:   1;
  1:   2,     1;
  2:   4,     6,     2;
  3:   8,    24,    18,      4;
  4:  16,    80,   108,     56,     10;
  5:  32,   240,   520,    460,    180,     26;
  6:  64,   672,  2160,   2920,   1920,    612,     76;
  7: 128,  1792,  8064,  15680,  15400,   8064,   2156,    232;
  8: 256,  4608, 27776,  74368, 102480,  78848,  34384,   7968,   764;
  9: 512, 11520, 89856, 319872, 594720, 632016, 399840, 149328, 30456, 2620;

The row polynomials satisfy P(n,x) = (2+n*x+(n-1)*x^2)*P(n-1,x) - (2*x^2+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[(2 + n*x + (n - 1)*x^2)*P[n - 1, x] - (2*x^2 + x^3)*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] = (2 + n*x + (n - 1)*x^2)*v[i - 1] - (2*x^2 + x^3)*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 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_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)

Related ?

Same fingerprint ?

• A122848 → 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