A108838 → D → (θ+1)²

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Triangle begins:
  0:   0;
  1:   2;
  2:   8,    16;
  3:  20,   120,     54;
  4:  40,   480,    648,     128;
  5:  70,  1400,   3780,    2240,     250;
  6: 112,  3360,  15120,   17920,    6000,     432;
  7: 168,  7056,  47628,   94080,   63000,   13608,    686;
  8: 240, 13440, 127008,  376320,  420000,  181440,  27440,  1024;
  9: 330, 23760, 299376, 1241856, 2079000, 1496880, 452760, 50688, 1458;

The row polynomials satisfy (4*n^2+8*n-12+(5*n^4+n^3-31*n^2+31*n-6)*x+(n^6-6*n^4+8*n^3-3*n^2)*x^2)*P(n,x) = (4*n^2+20*n+24+(5*n^4+16*n^3+35*n^2-28*n-52)*x+(n^6+3*n^5+21*n^4-15*n^3-42*n^2+32)*x^2+(3*n^6+3*n^5-6*n^4-9*n^3+9*n^2+4*n-4)*x^3)*P(n-1,x) + ((8*n+8)*x+(10*n^3+7*n^2-11*n-18)*x^2+(2*n^5+4*n^4-8*n^3-12*n^2+2*n+12)*x^3-(2*n^5+4*n^4+2*n^3-5*n^2-n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 20*n + 24 + (5*n^4 + 16*n^3 + 35*n^2 - 28*n - 52)*x + (n^6 + 3*n^5 + 21*n^4 - 15*n^3 - 42*n^2 + 32)*x^2 + (3*n^6 + 3*n^5 - 6*n^4 - 9*n^3 + 9*n^2 + 4*n - 4)*x^3)*P[n - 1, x] + ((8*n + 8)*x + (10*n^3 + 7*n^2 - 11*n - 18)*x^2 + (2*n^5 + 4*n^4 - 8*n^3 - 12*n^2 + 2*n + 12)*x^3 - (2*n^5 + 4*n^4 + 2*n^3 - 5*n^2 - n + 2)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 8*n - 12 + (5*n^4 + n^3 - 31*n^2 + 31*n - 6)*x + (n^6 - 6*n^4 + 8*n^3 - 3*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 20*n + 24 + (5*n^4 + 16*n^3 + 35*n^2 - 28*n - 52)*x + (n^6 + 3*n^5 + 21*n^4 - 15*n^3 - 42*n^2 + 32)*x^2 + (3*n^6 + 3*n^5 - 6*n^4 - 9*n^3 + 9*n^2 + 4*n - 4)*x^3)*v[i - 1] + ((8*n + 8)*x + (10*n^3 + 7*n^2 - 11*n - 18)*x^2 + (2*n^5 + 4*n^4 - 8*n^3 - 12*n^2 + 2*n + 12)*x^3 - (2*n^5 + 4*n^4 + 2*n^3 - 5*n^2 - n + 2)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 8*n - 12 + (5*n^4 + n^3 - 31*n^2 + 31*n - 6)*x + (n^6 - 6*n^4 + 8*n^3 - 3*n^2)*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 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 derivative of Triangle of Dyck paths counted by number of long interior inclines.?
    • Laguerre’s theorem on differential operators [SP14] 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) 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)
    • Laguerre’s theorem on differential operators [SP14] 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)

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