A086645 → basis A013611 → D

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Submission text ?

Triangle begins:
   1:       0;
   2:       4;
   3:      32,       32;
   4:     192,      576,       192;
   5:    1024,     5120,      6144,      1024;
   6:    5120,    35840,     76800,     51200,       5120;
   7:   24576,   221184,    688128,    860160,     368640,      24576;
   8:  114688,  1261568,   5160960,   9633792,    8028160,    2408448,    114688;
   9:  524288,  6815744,  34603008,  86507520,  110100480,   66060288,  14680064,   524288;
  10: 2359296, 35389440, 214695936, 674758656, 1167851520, 1089994752, 495452160, 84934656, 2359296;

The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (8*n^2-28*n+20+(24*n^2-92*n+68)*x)*P(n-1,x) - ((4*n-4)*x+(16*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 20 + (24*n^2 - 92*n + 68)*x)*P[n - 1, x] - ((4*n - 4)*x + (16*n - 16)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 20 + (24*n^2 - 92*n + 68)*x)*v[i - 1] - ((4*n - 4)*x + (16*n - 16)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    • Geometric basis map (elementary) polytool ? step ›

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

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

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

Related ?

Same fingerprint ?

• A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is th…
• A084534 → D
• A054142 → reverse
• A103327 → basis A013611
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
• A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
• A103327 → basis A013611 → reverse
• A121314: Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, …
• A085478 → reverse

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