Hadamard-factor parent of A155495 → D

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Submission text ?

Triangle begins:
   1:   6,     2;
   2:  15,    30,      3;
   3:  28,   140,     84,       4;
   4:  45,   420,    630,     180,       5;
   5:  66,   990,   2772,    1980,     330,       6;
   6:  91,  2002,   9009,   12012,    5005,     546,       7;
   7: 120,  3640,  24024,   51480,   40040,   10920,     840,      8;
   8: 153,  6120,  55692,  175032,  218790,  111384,   21420,   1224,     9;
   9: 190,  9690, 116280,  503880,  923780,  755820,  271320,  38760,  1710,   10;
  10: 231, 14630, 223839, 1279080, 3233230, 3879876, 2238390, 596904, 65835, 2310, 11;

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

P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (6*n^2 + n - 5)*x)*P[n - 1, x] + ((4*n + 4)*x - (4*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + 3*n + 1 + (6*n^2 + n - 5)*x)*v[i - 1] + ((4*n + 4)*x - (4*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - n)); 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    • 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
• A086645 → D
• A034870 → Veronese 2 → D
• A103327 → reverse
• A091042 → reverse
• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

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