A091044 → basis A013609 → D+1

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Triangle begins:
  0:       1;
  1:       8,        4;
  2:      48,       56,       12;
  3:     256,      512,      256,        32;
  4:    1280,     3712,     3264,       960,        80;
  5:    6144,    23552,    30720,     16128,      3200,      192;
  6:   28672,   137216,   240640,    190464,     68096,     9856,      448;
  7:  131072,   753664,  1671168,   1802240,    983040,   258048,    28672,    1024;
  8:  589824,  3964928, 10665984,  14696448,  11038720,  4460544,   903168,   79872,   2304;
  9: 2621440, 20185088, 63963136, 107806720, 104366080, 58572800, 18382848, 2973696, 215040, 5120;

The row polynomials satisfy (2*n^3-n^2+(4*n^2-4*n)*x+2*n*x^2)*P(n,x) = (8*n^3+4*n^2-4*n+(12*n^3+14*n^2-22*n)*x+(24*n^2-12*n+4)*x^2+(12*n-4)*x^3)*P(n-1,x) - ((4*n^2+4*n)*x+(8*n^2+16*n)*x^2+(16*n+4)*x^3+8*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[((8*n^3 + 4*n^2 - 4*n + (12*n^3 + 14*n^2 - 22*n)*x + (24*n^2 - 12*n + 4)*x^2 + (12*n - 4)*x^3)*P[n - 1, x] - ((4*n^2 + 4*n)*x + (8*n^2 + 16*n)*x^2 + (16*n + 4)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - n^2 + (4*n^2 - 4*n)*x + 2*n*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] = ((8*n^3 + 4*n^2 - 4*n + (12*n^3 + 14*n^2 - 22*n)*x + (24*n^2 - 12*n + 4)*x^2 + (12*n - 4)*x^3)*v[i - 1] - ((4*n^2 + 4*n)*x + (8*n^2 + 16*n)*x^2 + (16*n + 4)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - n^2 + (4*n^2 - 4*n)*x + 2*n*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 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    • Geometric basis map (elementary) polytool ? step ›

      • from One half of odd-numbered entries of even-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 One half of odd-numbered entries of even-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 One half of odd-numbered entries of even-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)
    • Geometric basis map (elementary) polytool ? step ›

      • from One half of odd-numbered entries of even-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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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