A091042 → basis A013610 → D

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Submission text ?

Triangle begins:
   1:       0;
   2:       9;
   3:      60,       90;
   4:     336,     1008,       567;
   5:    1728,     7776,      9720,      2916;
   6:    8448,    50688,     99792,     71280,     13365;
   7:   39936,   299520,    808704,    943488,    442260,     56862;
   8:  184320,  1658880,   5702400,   9331200,   7348320,   2449440,    229635;
   9:  835584,  8773632,  36661248,  77552640,  87246720,  49968576,  12492144,   892296;
  10: 3735552, 44826624, 220631040, 573640704, 845095680, 702079488, 307159776, 59836320, 3365793;

The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (8*n^2-20*n+8+(18*n^2-51*n+21)*x)*P(n-1,x) - ((4*n-2)*x+(12*n-6)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 9.

P[1, x_] := 0;
P[2, x_] := 9;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 20*n + 8 + (18*n^2 - 51*n + 21)*x)*P[n - 1, x] - ((4*n - 2)*x + (12*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 9; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 20*n + 8 + (18*n^2 - 51*n + 21)*x)*v[i - 1] - ((4*n - 2)*x + (12*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A091044 → basis A013610
• A091044 → basis A013610 → reverse
• A034870 → Veronese 2 (j=1) → basis A013610

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023