A086645 → D → θ+1

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Submission text ?

Triangle begins:
   1:   0;
   2:   1;
   3:   6,     4;
   4:  15,    60,      9;
   5:  28,   280,    252,     16;
   6:  45,   840,   1890,    720,      25;
   7:  66,  1980,   8316,   7920,    1650,     36;
   8:  91,  4004,  27027,  48048,   25025,   3276,     49;
   9: 120,  7280,  72072, 205920,  200200,  65520,   5880,   64;
  10: 153, 12240, 167076, 700128, 1093950, 668304, 149940, 9792, 81;

The row polynomials satisfy -(2*n^2-9*n+10-(4*n^4-36*n^3+121*n^2-180*n+100)*x)*P(n,x) = -(2*n^2-5*n+3-(4*n^4-28*n^3+59*n^2-31*n-4)*x-(12*n^4-84*n^3+203*n^2-198*n+67)*x^2)*P(n-1,x) - ((4*n-4)*x-(8*n^3-32*n^2+42*n-18)*x^2+(8*n^3-32*n^2+38*n-14)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 3 - (4*n^4 - 28*n^3 + 59*n^2 - 31*n - 4)*x - (12*n^4 - 84*n^3 + 203*n^2 - 198*n + 67)*x^2)*P[n - 1, x] - ((4*n - 4)*x - (8*n^3 - 32*n^2 + 42*n - 18)*x^2 + (8*n^3 - 32*n^2 + 38*n - 14)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + 9*n - 10 + (4*n^4 - 36*n^3 + 121*n^2 - 180*n + 100)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 5*n + 3 - (4*n^4 - 28*n^3 + 59*n^2 - 31*n - 4)*x - (12*n^4 - 84*n^3 + 203*n^2 - 198*n + 67)*x^2)*v[i - 1] - ((4*n - 4)*x - (8*n^3 - 32*n^2 + 42*n - 18)*x^2 + (8*n^3 - 32*n^2 + 38*n - 14)*x^3)*deriv(v[i - 1], x))/(-2*n^2 + 9*n - 10 + (4*n^4 - 36*n^3 + 121*n^2 - 180*n + 100)*x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from derivative of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    • Laguerre’s theorem on differential operators [SP14] 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)
    • Laguerre’s theorem on differential operators [SP14] 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 ?

• A103327 → θ+1
• A091042 → reverse → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023