A091042 → reverse → (θ+1)²

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Submission text ?

Triangle begins:
   1:  1;
   2:  3,    4;
   3:  5,   40,      9;
   4:  7,  140,    189,     16;
   5:  9,  336,   1134,    576,      25;
   6: 11,  660,   4158,   5280,    1375,      36;
   7: 13, 1144,  11583,  27456,   17875,    2808,      49;
   8: 15, 1820,  27027, 102960,  125125,   49140,    5145,     64;
   9: 17, 2720,  55692, 311168,  607750,  445536,  116620,   8704,    81;
  10: 19, 3876, 104652, 806208, 2309450, 2720952, 1329468, 248064, 13851, 100;

The row polynomials satisfy (2*n-3-(20*n^3-88*n^2+127*n-60)*x+(8*n^5-52*n^4+134*n^3-171*n^2+108*n-27)*x^2)*P(n,x) = (2*n-1-(20*n^3-68*n^2+45*n+24)*x+(8*n^5-44*n^4-14*n^3+275*n^2-283*n+21)*x^2+(24*n^5-100*n^4+122*n^3-19*n^2-34*n+4)*x^3)*P(n-1,x) + (4*x-(40*n^2-56*n+6)*x^2+(16*n^4-32*n^3+44*n^2-40*n)*x^3-(16*n^4-32*n^3+4*n^2+16*n-2)*x^4)*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 - (20*n^3 - 68*n^2 + 45*n + 24)*x + (8*n^5 - 44*n^4 - 14*n^3 + 275*n^2 - 283*n + 21)*x^2 + (24*n^5 - 100*n^4 + 122*n^3 - 19*n^2 - 34*n + 4)*x^3)*P[n - 1, x] + (4*x - (40*n^2 - 56*n + 6)*x^2 + (16*n^4 - 32*n^3 + 44*n^2 - 40*n)*x^3 - (16*n^4 - 32*n^3 + 4*n^2 + 16*n - 2)*x^4)*D[P[n - 1, x], x])/(2*n - 3 - (20*n^3 - 88*n^2 + 127*n - 60)*x + (8*n^5 - 52*n^4 + 134*n^3 - 171*n^2 + 108*n - 27)*x^2)]];
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 - (20*n^3 - 68*n^2 + 45*n + 24)*x + (8*n^5 - 44*n^4 - 14*n^3 + 275*n^2 - 283*n + 21)*x^2 + (24*n^5 - 100*n^4 + 122*n^3 - 19*n^2 - 34*n + 4)*x^3)*v[i - 1] + (4*x - (40*n^2 - 56*n + 6)*x^2 + (16*n^4 - 32*n^3 + 44*n^2 - 40*n)*x^3 - (16*n^4 - 32*n^3 + 4*n^2 + 16*n - 2)*x^4)*deriv(v[i - 1], x))/(2*n - 3 - (20*n^3 - 88*n^2 + 127*n - 60)*x + (8*n^5 - 52*n^4 + 134*n^3 - 171*n^2 + 108*n - 27)*x^2)); 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 ?

• A103327 → (θ+1)²
• A086645 → D → (θ+1)²

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