A086645 → γ → D+1

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Submission text ?

Triangle begins:
   1:   1;
   2:   1;
   3:   5,    4;
   4:  13,   12;
   5:  25,   56,    16;
   6:  41,  200,    80;
   7:  61,  540,   432,    64;
   8:  85, 1204,  1904,   448;
   9: 113, 2352,  6496,  2816,  256;
  10: 145, 4176, 18144, 14592, 2304;

The row polynomials satisfy (2*n^3-14*n^2+33*n-26+(8*n^2-46*n+60)*x+(8*n-16)*x^2)*P(n,x) = (2*n^3-10*n^2+17*n-10+(8*n^3-48*n^2+62*n+4)*x+(32*n^2-144*n+160)*x^2+(32*n-64)*x^3)*P(n-1,x) + ((4*n^2-12*n+6)*x-(16*n^2-64*n+52)*x^2-(64*n-128)*x^3-64*x^4)*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[((2*n^3 - 10*n^2 + 17*n - 10 + (8*n^3 - 48*n^2 + 62*n + 4)*x + (32*n^2 - 144*n + 160)*x^2 + (32*n - 64)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 6)*x - (16*n^2 - 64*n + 52)*x^2 - (64*n - 128)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 14*n^2 + 33*n - 26 + (8*n^2 - 46*n + 60)*x + (8*n - 16)*x^2)]];
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] = ((2*n^3 - 10*n^2 + 17*n - 10 + (8*n^3 - 48*n^2 + 62*n + 4)*x + (32*n^2 - 144*n + 160)*x^2 + (32*n - 64)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 6)*x - (16*n^2 - 64*n + 52)*x^2 - (64*n - 128)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 14*n^2 + 33*n - 26 + (8*n^2 - 46*n + 60)*x + (8*n - 16)*x^2)); 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → D+1

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