A091042 → basis A007318 → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:      1;
   2:      4,       6;
   3:     16,      40,      15;
   4:     64,     224,     168,       28;
   5:    256,    1152,    1296,      480,      45;
   6:   1024,    5632,    8448,     4928,    1100,      66;
   7:   4096,   26624,   49920,    39936,   14560,    2184,      91;
   8:  16384,  122880,  276480,   281600,  144000,   36288,    3920,    120;
   9:  65536,  557056, 1462272,  1810432, 1196800,  430848,   79968,   6528,   153;
  10: 262144, 2490368, 7471104, 10895360, 8852480, 4173312, 1123584, 160512, 10260, 190;

The row polynomials satisfy (8*n^2-20*n+12+(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = (32*n^2-80*n+48+(16*n^3-76*n+60)*x+(12*n^3-20*n^2-5*n+12)*x^2)*P(n-1,x) - ((16*n-16)*x+(8*n^2+16*n-22)*x^2+(8*n^2-6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 80*n + 48 + (16*n^3 - 76*n + 60)*x + (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*P[n - 1, x] - ((16*n - 16)*x + (8*n^2 + 16*n - 22)*x^2 + (8*n^2 - 6)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 20*n + 12 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 80*n + 48 + (16*n^3 - 76*n + 60)*x + (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*v[i - 1] - ((16*n - 16)*x + (8*n^2 + 16*n - 22)*x^2 + (8*n^2 - 6)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 20*n + 12 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by 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 basis_map_A007318 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • 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 reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (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