A086645 → γ → θ+1

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Submission text ?

Triangle begins:
   1: 1;
   2: 1;
   3: 1,   8;
   4: 1,  24;
   5: 1,  48,   48;
   6: 1,  80,  240;
   7: 1, 120,  720,   256;
   8: 1, 168, 1680,  1792;
   9: 1, 224, 3360,  7168,  1280;
  10: 1, 288, 6048, 21504, 11520;

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

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 - (2*n^3 - 6*n^2 - 20*n + 48)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*P[n - 1, x] - (6*x - (4*n^2 + 4*n + 24)*x^2 + (16*n^2 + 16*n)*x^3)*D[P[n - 1, x], x])/(-3*n + 6 + (2*n^3 - 6*n^2 + 4*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(3*n - 6 - (2*n^3 - 6*n^2 - 20*n + 48)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*v[i - 1] - (6*x - (4*n^2 + 4*n + 24)*x^2 + (16*n^2 + 16*n)*x^3)*deriv(v[i - 1], x))/(-3*n + 6 + (2*n^3 - 6*n^2 + 4*n)*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 gamma_polynomial of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? 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)

Related ?

Same fingerprint ?

• A013609 → Veronese 2 → 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