A111125 → reverse → D

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Submission text ?

Triangle begins:
   1:  0;
   2:  3;
   3:  5,  10;
   4:  7,  28,   21;
   5:  9,  54,   90,   36;
   6: 11,  88,  231,  220,    55;
   7: 13, 130,  468,  728,   455,    78;
   8: 15, 180,  825, 1800,  1890,   840,  105;
   9: 17, 238, 1326, 3740,  5610,  4284, 1428,  136;
  10: 19, 304, 1995, 6916, 13585, 15048, 8778, 2280, 171;

The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (4*n^2-10*n+4+(12*n^2-34*n+14)*x)*P(n-1,x) - ((2*n-1)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 4 + (12*n^2 - 34*n + 14)*x)*P[n - 1, x] - ((2*n - 1)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 10*n + 4 + (12*n^2 - 34*n + 14)*x)*v[i - 1] - ((2*n - 1)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12)); 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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
        • 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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
                • ↑ 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 ?

• A091044 → basis A013611 → reverse
• A172431 → reverse
• A172431: Even row Pascal-square read by antidiagonals.
• A034870 → Veronese 2 (j=1) → basis A013611
• A082985 → D
• A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• A078812 → reverse
• A091044 → basis A013611
• A128908: Riordan array (1, x/(1-x)^2).

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