A054142 → reverse → D+1

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Submission text ?

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  4,   5,    1;
  3:  7,  16,    8,    1;
  4: 11,  40,   36,   11,    1;
  5: 16,  85,  119,   64,   14,    1;
  6: 22, 161,  322,  264,  100,   17,    1;
  7: 29, 280,  756,  870,  495,  144,   20,   1;
  8: 37, 456, 1596, 2442, 1925,  832,  196,  23,  1;
  9: 46, 705, 3102, 6072, 6292, 3731, 1295, 256, 26, 1;

The row polynomials satisfy (2*n^3-3*n^2+5*n-2-(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (2*n^3+n^2+3*n-2+(n^3+n^2-6*n+2)*x-(5*n-1)*x^2-n*x^3)*P(n-1,x) + ((4*n^2-8)*x+(n^2-18)*x^2-8*x^3-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[((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*P[n - 1, x] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*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] = ((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*v[i - 1] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 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 [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) (above)

Related ?

Same fingerprint ?

• A165253 → 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