A100861 → D → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:   0;
  1:   0;
  2:   0;
  3:   0;
  4:   6;
  5:  30;
  6:  90,   90;
  7: 210,  630;
  8: 420, 2520,  1260;
  9: 756, 7560, 11340;

The row polynomials satisfy (n-4)*P(n,x) = (n+(n^2-5*n)*x)*P(n-1,x) - 2*n*x^2*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 6.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 0;
P[4, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((n + (n^2 - 5*n)*x)*P[n - 1, x] - 2*n*x^2*D[P[n - 1, x], x])/(n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 0; v[5] = 6; for(n = 5, N, my(i = n + 1); v[i] = ((n + (n^2 - 5*n)*x)*v[i - 1] - 2*n*x^2*deriv(v[i - 1], x))/(n - 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 derivative of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).?
    • Laguerre’s theorem on differential operators [SP14] 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)).?
            • 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 (above)
                • 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 [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
                • 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)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Laguerre’s theorem on differential operators [SP14] 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 (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
• A100861 → D
• A049403 → reverse
• A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).
• A122848: Exponential Riordan array (1, x(1+x/2)).

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.