A054142 → D → θ+1

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Triangle begins:
  0:  0;
  1:  1;
  2:  3,   4;
  3:  5,  24,    9;
  4:  7,  60,   90,    16;
  5:  9, 112,  315,   240,    25;
  6: 11, 180,  756,  1120,   525,    36;
  7: 13, 264, 1485,  3360,  3150,  1008,    49;
  8: 15, 364, 2574,  7920, 11550,  7560,  1764,   64;
  9: 17, 480, 4095, 16016, 32175, 33264, 16170, 2880, 81;

The row polynomials satisfy (8*n^3-24*n^2+22*n-6+(16*n^4-80*n^3+148*n^2-120*n+36)*x)*P(n,x) = (8*n^3-16*n^2+10*n-2+(16*n^4-16*n^3-36*n^2+28*n-4)*x+(48*n^4-160*n^3+148*n^2-40*n+16)*x^2)*P(n-1,x) - ((4*n^2-4*n+1)*x+(8*n^3+4*n^2-12*n+2)*x^2+(32*n^3-48*n^2+16*n-8)*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[((8*n^3 - 16*n^2 + 10*n - 2 + (16*n^4 - 16*n^3 - 36*n^2 + 28*n - 4)*x + (48*n^4 - 160*n^3 + 148*n^2 - 40*n + 16)*x^2)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (8*n^3 + 4*n^2 - 12*n + 2)*x^2 + (32*n^3 - 48*n^2 + 16*n - 8)*x^3)*D[P[n - 1, x], x])/(8*n^3 - 24*n^2 + 22*n - 6 + (16*n^4 - 80*n^3 + 148*n^2 - 120*n + 36)*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] = ((8*n^3 - 16*n^2 + 10*n - 2 + (16*n^4 - 16*n^3 - 36*n^2 + 28*n - 4)*x + (48*n^4 - 160*n^3 + 148*n^2 - 40*n + 16)*x^2)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (8*n^3 + 4*n^2 - 12*n + 2)*x^2 + (32*n^3 - 48*n^2 + 16*n - 8)*x^3)*deriv(v[i - 1], x))/(8*n^3 - 24*n^2 + 22*n - 6 + (16*n^4 - 80*n^3 + 148*n^2 - 120*n + 36)*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 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
        • 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
        • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

          • from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
            • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                no recorded property covers this requirement
            • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                no recorded property covers this requirement
            • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                no recorded property covers this requirement
            • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                no recorded property covers this requirement
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Gamma criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Gamma criterion (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangular array binomial(2*n-k, k), k=0..n, n >= 0.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Gamma 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