A108838 → reverse → (θ+1)²

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Submission text ?

Triangle begins:
  0: 2;
  1: 2,  12;
  2: 2,  32,    36;
  3: 2,  60,   180,     80;
  4: 2,  96,   540,    640,    150;
  5: 2, 140,  1260,   2800,   1750,    252;
  6: 2, 192,  2520,   8960,  10500,   4032,    392;
  7: 2, 252,  4536,  23520,  44100,  31752,   8232,    576;
  8: 2, 320,  7560,  53760, 147000, 169344,  82320,  15360,   810;
  9: 2, 396, 11880, 110880, 415800, 698544, 543312, 190080, 26730, 1100;

The row polynomials satisfy -(n^2+2*n-(n^4+4*n^3+4*n^2)*x)*P(n,x) = -(n^2+2*n-(n^4+4*n^3-2*n^2-8*n-4)*x-(3*n^4+14*n^3+19*n^2+14*n+4)*x^2)*P(n-1,x) - ((2*n+2)*x-(2*n^3+9*n^2+12*n+6)*x^2+(2*n^3+9*n^2+10*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 2*n - (n^4 + 4*n^3 - 2*n^2 - 8*n - 4)*x - (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*P[n - 1, x] - ((2*n + 2)*x - (2*n^3 + 9*n^2 + 12*n + 6)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^2 - 2*n + (n^4 + 4*n^3 + 4*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 2*n - (n^4 + 4*n^3 - 2*n^2 - 8*n - 4)*x - (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*v[i - 1] - ((2*n + 2)*x - (2*n^3 + 9*n^2 + 12*n + 6)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^2 - 2*n + (n^4 + 4*n^3 + 4*n^2)*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  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 Dyck paths counted by number of long interior inclines.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of Dyck paths counted by number of long interior inclines.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle of Dyck paths counted by number of long interior inclines.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of Dyck paths counted by number of long interior inclines.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A062196 → θ+1
• A062196 → 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