A082985 → reverse → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  1;
  1:  3,    4;
  2:  5,   20,     9;
  3:  7,   56,    63,    16;
  4:  9,  120,   243,   144,    25;
  5: 11,  220,   693,   704,   275,    36;
  6: 13,  364,  1638,  2496,  1625,   468,    49;
  7: 15,  560,  3402,  7200,  6875,  3240,   735,   64;
  8: 17,  816,  6426, 17952, 23375, 15912,  5831, 1088,   81;
  9: 19, 1140, 11286, 40128, 67925, 62244, 32585, 9728, 1539, 100;

The row polynomials satisfy (8*n-4-(20*n^3-30*n^2+14*n-2)*x+(2*n^5-5*n^4+4*n^3-n^2)*x^2)*P(n,x) = (8*n+4-(20*n^3-10*n^2-22*n+2)*x+(2*n^5-3*n^4-18*n^3+n^2+10*n)*x^2+(n^5-2*n^3+n)*x^3)*P(n-1,x) + (16*x-(40*n^2-28)*x^2+(4*n^4-18*n^2+10)*x^3+(n^4-2*n^2+1)*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 + 4 - (20*n^3 - 10*n^2 - 22*n + 2)*x + (2*n^5 - 3*n^4 - 18*n^3 + n^2 + 10*n)*x^2 + (n^5 - 2*n^3 + n)*x^3)*P[n - 1, x] + (16*x - (40*n^2 - 28)*x^2 + (4*n^4 - 18*n^2 + 10)*x^3 + (n^4 - 2*n^2 + 1)*x^4)*D[P[n - 1, x], x])/(8*n - 4 - (20*n^3 - 30*n^2 + 14*n - 2)*x + (2*n^5 - 5*n^4 + 4*n^3 - 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] = ((8*n + 4 - (20*n^3 - 10*n^2 - 22*n + 2)*x + (2*n^5 - 3*n^4 - 18*n^3 + n^2 + 10*n)*x^2 + (n^5 - 2*n^3 + n)*x^3)*v[i - 1] + (16*x - (40*n^2 - 28)*x^2 + (4*n^4 - 18*n^2 + 10)*x^3 + (n^4 - 2*n^2 + 1)*x^4)*deriv(v[i - 1], x))/(8*n - 4 - (20*n^3 - 30*n^2 + 14*n - 2)*x + (2*n^5 - 5*n^4 + 4*n^3 - 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
        • 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 reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from shift_x_plus_one of Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q…?
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…?
                    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A111125 → (θ+1)²

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