A103327 → D → θ+1

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Triangle begins:
  0:   0;
  1:   1;
  2:  10,     4;
  3:  35,    84,      9;
  4:  84,   504,    324,      16;
  5: 165,  1848,   2970,     880,      25;
  6: 286,  5148,  15444,   11440,    1950,     36;
  7: 455, 12012,  57915,   80080,   34125,   3780,     49;
  8: 680, 24752, 175032,  388960,  309400,  85680,   6664,    64;
  9: 969, 46512, 453492, 1478048, 1889550, 976752, 189924, 10944, 81;

The row polynomials satisfy (2*n^2-5*n+3+(4*n^4-20*n^3+37*n^2-30*n+9)*x)*P(n,x) = (2*n^2+n+(4*n^4-8*n^3+13*n^2-23*n+8)*x+(12*n^4-40*n^3+37*n^2-10*n+4)*x^2)*P(n-1,x) + ((4*n-2)*x+(8*n^3-12*n^2)*x^2-(8*n^3-12*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (4*n^4 - 8*n^3 + 13*n^2 - 23*n + 8)*x + (12*n^4 - 40*n^3 + 37*n^2 - 10*n + 4)*x^2)*P[n - 1, x] + ((4*n - 2)*x + (8*n^3 - 12*n^2)*x^2 - (8*n^3 - 12*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3 + (4*n^4 - 20*n^3 + 37*n^2 - 30*n + 9)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n + (4*n^4 - 8*n^3 + 13*n^2 - 23*n + 8)*x + (12*n^4 - 40*n^3 + 37*n^2 - 10*n + 4)*x^2)*v[i - 1] + ((4*n - 2)*x + (8*n^3 - 12*n^2)*x^2 - (8*n^3 - 12*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3 + (4*n^4 - 20*n^3 + 37*n^2 - 30*n + 9)*x)); 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 derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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