A367270 → D → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  0;
  1:  0;
  2:  2;
  3:  6,   12;
  4: 12,   72,    36;
  5: 20,  240,   360,     80;
  6: 30,  600,  1800,   1200,    150;
  7: 42, 1260,  6300,   8400,   3150,   252;
  8: 56, 2352, 17640,  39200,  29400,  7056,   392;
  9: 72, 4032, 42336, 141120, 176400, 84672, 14112, 576;

The row polynomials satisfy (n^3-5*n^2+8*n-4)*P(n,x) = (n^3-3*n^2+2*n+(3*n^3-11*n^2+10*n)*x)*P(n-1,x) + ((2*n^2-3*n)*x-(2*n^2-3*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 2*n + (3*n^3 - 11*n^2 + 10*n)*x)*P[n - 1, x] + ((2*n^2 - 3*n)*x - (2*n^2 - 3*n)*x^2)*D[P[n - 1, x], x])/(n^3 - 5*n^2 + 8*n - 4)]];
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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 2*n + (3*n^3 - 11*n^2 + 10*n)*x)*v[i - 1] + ((2*n^2 - 3*n)*x - (2*n^2 - 3*n)*x^2)*deriv(v[i - 1], x))/(n^3 - 5*n^2 + 8*n - 4)); 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 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 ?

• A001263 → θ+1 → reverse
• narayana → θ+1
• A001263 → θ+1
• A069466 → D
• A008459 → D
• narayana → θ+1 → reverse
• A367270: Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - 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