A108838 → reverse → D

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Submission text ?

Triangle begins:
  0:  0;
  1:  3;
  2:  8,    8;
  3: 15,   40,    15;
  4: 24,  120,   120,    24;
  5: 35,  280,   525,   280,    35;
  6: 48,  560,  1680,  1680,   560,    48;
  7: 63, 1008,  4410,  7056,  4410,  1008,    63;
  8: 80, 1680, 10080, 23520, 23520, 10080,  1680,   80;
  9: 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99;

The row polynomials satisfy (n^3+3*n^2-n-3)*P(n,x) = (n^3+5*n^2+6*n+(3*n^3+5*n^2-2*n)*x)*P(n-1,x) + ((2*n^2+4*n)*x-(2*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 5*n^2 + 6*n + (3*n^3 + 5*n^2 - 2*n)*x)*P[n - 1, x] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 5*n^2 + 6*n + (3*n^3 + 5*n^2 - 2*n)*x)*v[i - 1] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - n - 3)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → D → D
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 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