A089627 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
   1:    0;
   2:    0;
   3:    2;
   4:    6;
   5:   24,    12;
   6:   80,    60;
   7:  270,   300,    60;
   8:  882,  1260,   420;
   9: 2856,  5040,  2520,  280;
  10: 9144, 19152, 12600, 2520;

The row polynomials satisfy (n-3)*P(n,x) = (5*n-17+(4*n-16)*x)*P(n-1,x) - (6+14*x+8*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 17 + (4*n - 16)*x)*P[n - 1, x] - (6 + 14*x + 8*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((5*n - 17 + (4*n - 16)*x)*v[i - 1] - (6 + 14*x + 8*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.?
        • Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
            • Parent-row provers of polytool polytool ? step › direct check
            • Parent-row provers of polytool polytool ? 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 multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from gamma_polynomial of narayana?
            • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

              • from narayana?
                • 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
                • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from narayana?
                    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                    • ↑ Three-term recurrence criterion (above)
                    • ↑ Liu–Wang theorem for three-term recurrences (above)
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from narayana?
                        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                        • ↑ Three-term recurrence criterion (above)
                        • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.?
        • ↑ Pólya–Schur theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.?
        • ↑ Pólya–Schur theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A391403 → x ↦ x+1 → D
• A132812 → γ → x ↦ x+1
• narayana → γ → basis A007318
• A107131 → x ↦ x/(x+1)
• A055151 → basis A007318

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