A103327 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:       1;
  1:       6,        2;
  2:      40,       32,        4;
  3:     224,      352,      120,        8;
  4:    1152,     2816,     1920,      384,       16;
  5:    5632,    18944,    20608,     8320,     1120,       32;
  6:   26624,   114688,   175104,   114688,    31360,     3072,      64;
  7:  122880,   647168,  1284096,  1198080,   537600,   107520,    8064,     128;
  8:  557056,  3473408,  8519680, 10452992,  6758400,  2236416,  344064,   20480,   256;
  9: 2490368, 17956864, 52592640, 80510976, 69115904, 33251328, 8515584, 1044480, 50688, 512;

The row polynomials satisfy (4*n^3-8*n^2+5*n-1+(8*n^2-16*n+6)*x+(4*n-2)*x^2)*P(n,x) = (16*n^3-16*n^2-4*n+4+(24*n^3-8*n^2-54*n+26)*x+(48*n^2-72*n+32)*x^2+(24*n-20)*x^3)*P(n-1,x) - ((8*n^2-2)*x+(16*n^2+16*n-12)*x^2+(32*n-8)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 16*n^2 - 4*n + 4 + (24*n^3 - 8*n^2 - 54*n + 26)*x + (48*n^2 - 72*n + 32)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - ((8*n^2 - 2)*x + (16*n^2 + 16*n - 12)*x^2 + (32*n - 8)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 8*n^2 + 5*n - 1 + (8*n^2 - 16*n + 6)*x + (4*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 - 16*n^2 - 4*n + 4 + (24*n^3 - 8*n^2 - 54*n + 26)*x + (48*n^2 - 72*n + 32)*x^2 + (24*n - 20)*x^3)*v[i - 1] - ((8*n^2 - 2)*x + (16*n^2 + 16*n - 12)*x^2 + (32*n - 8)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 8*n^2 + 5*n - 1 + (8*n^2 - 16*n + 6)*x + (4*n - 2)*x^2)); 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    • Geometric basis map (elementary) polytool ? step ›

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

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

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

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

Related ?

Same fingerprint ?

• A123519 → reverse → D+1

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