A367270 → D → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    0;
  1:    0;
  2:    0;
  3:    6;
  4:   36,    24;
  5:  120,   240,     60;
  6:  300,  1200,    900,    120;
  7:  630,  4200,   6300,   2520,   210;
  8: 1176, 11760,  29400,  23520,  5880,   336;
  9: 2016, 28224, 105840, 141120, 70560, 12096, 504;

The row polynomials satisfy (n^2-5*n+6)*P(n,x) = (n^2-n+(3*n^2-13*n+12)*x)*P(n-1,x) + ((2*n-3)*x-(2*n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 6.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (3*n^2 - 13*n + 12)*x)*P[n - 1, x] + ((2*n - 3)*x - (2*n - 3)*x^2)*D[P[n - 1, x], x])/(n^2 - 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 6; for(n = 4, N, my(i = n + 1); v[i] = ((n^2 - n + (3*n^2 - 13*n + 12)*x)*v[i - 1] + ((2*n - 3)*x - (2*n - 3)*x^2)*deriv(v[i - 1], x))/(n^2 - 5*n + 6)); 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • 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
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A108838 → reverse
• A108838: Triangle of Dyck paths counted by number of long interior inclines.
• A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).
• narayana → D
• A001263 → D
• A132812 → D
• A281260: Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= …
• A281260 → reverse
• A352880 → reverse
• A062196 → D

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