A078812 → reverse → D

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Submission text ?

Triangle begins:
   1:  0;
   2:  2;
   3:  4,   6;
   4:  6,  20,   12;
   5:  8,  42,   60,   20;
   6: 10,  72,  168,  140,    30;
   7: 12, 110,  360,  504,   280,    42;
   8: 14, 156,  660, 1320,  1260,   504,   56;
   9: 16, 210, 1092, 2860,  3960,  2772,  840,   72;
  10: 18, 272, 1680, 5460, 10010, 10296, 5544, 1320, 90;

The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2-5*n+3+(6*n^2-19*n+12)*x)*P(n-1,x) - ((n-1)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 + (6*n^2 - 19*n + 12)*x)*P[n - 1, x] - ((n - 1)*x + (4*n - 4)*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] = 2; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 + (6*n^2 - 19*n + 12)*x)*v[i - 1] - ((n - 1)*x + (4*n - 4)*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

Proof chains ? JSON

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

  • from reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A085478 → D
• A091044 → basis A013611 → D
• A172431 → D
• A054142 → reverse → D

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