A091042 → reverse → θ+1

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Submission text ?

Triangle begins:
   1:  1;
   2:  3,    2;
   3:  5,   20,     3;
   4:  7,   70,    63,      4;
   5:  9,  168,   378,    144,      5;
   6: 11,  330,  1386,   1320,    275,      6;
   7: 13,  572,  3861,   6864,   3575,    468,      7;
   8: 15,  910,  9009,  25740,  25025,   8190,    735,     8;
   9: 17, 1360, 18564,  77792, 121550,  74256,  16660,  1088,    9;
  10: 19, 1938, 34884, 201552, 461890, 453492, 189924, 31008, 1539, 10;

The row polynomials satisfy -(2*n-3-(4*n^3-16*n^2+21*n-9)*x)*P(n,x) = -(2*n-1-(4*n^3-12*n^2-n+19)*x-(12*n^3-36*n^2+23*n+4)*x^2)*P(n-1,x) - (4*x-(8*n^2-8*n+2)*x^2+(8*n^2-8*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 - (4*n^3 - 12*n^2 - n + 19)*x - (12*n^3 - 36*n^2 + 23*n + 4)*x^2)*P[n - 1, x] - (4*x - (8*n^2 - 8*n + 2)*x^2 + (8*n^2 - 8*n - 2)*x^3)*D[P[n - 1, x], x])/(-2*n + 3 + (4*n^3 - 16*n^2 + 21*n - 9)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 1 - (4*n^3 - 12*n^2 - n + 19)*x - (12*n^3 - 36*n^2 + 23*n + 4)*x^2)*v[i - 1] - (4*x - (8*n^2 - 8*n + 2)*x^2 + (8*n^2 - 8*n - 2)*x^3)*deriv(v[i - 1], x))/(-2*n + 3 + (4*n^3 - 16*n^2 + 21*n - 9)*x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by 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 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • 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 reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A086645 → D → θ+1
• A103327 → 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