A054142 → reverse → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  6,   3;
  3: 1, 12,  15,    4;
  4: 1, 20,  45,   28,    5;
  5: 1, 30, 105,  112,   45,    6;
  6: 1, 42, 210,  336,  225,   66,    7;
  7: 1, 56, 378,  840,  825,  396,   91,   8;
  8: 1, 72, 630, 1848, 2475, 1716,  637, 120,   9;
  9: 1, 90, 990, 3696, 6435, 6006, 3185, 960, 153, 10;

The row polynomials satisfy -(12*n-6-(2*n^3-3*n^2+n)*x)*P(n,x) = -(12*n-6-(2*n^3-3*n^2-11*n)*x-(n^3-n)*x^2)*P(n-1,x) - (24*x-(4*n^2-10)*x^2-(n^2-1)*x^3)*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[(-(12*n - 6 - (2*n^3 - 3*n^2 - 11*n)*x - (n^3 - n)*x^2)*P[n - 1, x] - (24*x - (4*n^2 - 10)*x^2 - (n^2 - 1)*x^3)*D[P[n - 1, x], x])/(-12*n + 6 + (2*n^3 - 3*n^2 + n)*x)]];
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] = (-(12*n - 6 - (2*n^3 - 3*n^2 - 11*n)*x - (n^3 - n)*x^2)*v[i - 1] - (24*x - (4*n^2 - 10)*x^2 - (n^2 - 1)*x^3)*deriv(v[i - 1], x))/(-12*n + 6 + (2*n^3 - 3*n^2 + n)*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 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 → 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