A121314 → D

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Submission text ?

Triangle begins:
  0: 0;
  1: 1;
  2: 1,  2;
  3: 1,  6,   3;
  4: 1, 10,  18,    4;
  5: 1, 14,  45,   40,    5;
  6: 1, 18,  84,  140,   75,    6;
  7: 1, 22, 135,  336,  350,  126,    7;
  8: 1, 26, 198,  660, 1050,  756,  196,   8;
  9: 1, 30, 273, 1144, 2475, 2772, 1470, 288, 9;

The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-32*n^2+42*n-18)*x)*P(n,x) = (4*n^2-10*n+6+(8*n^3-8*n^2-26*n+26)*x+(24*n^3-72*n^2+46*n+8)*x^2)*P(n-1,x) - ((2*n-2)*x+(4*n^2+4*n-9)*x^2+(16*n^2-16*n-4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (8*n^3 - 8*n^2 - 26*n + 26)*x + (24*n^3 - 72*n^2 + 46*n + 8)*x^2)*P[n - 1, x] - ((2*n - 2)*x + (4*n^2 + 4*n - 9)*x^2 + (16*n^2 - 16*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 32*n^2 + 42*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 10*n + 6 + (8*n^3 - 8*n^2 - 26*n + 26)*x + (24*n^3 - 72*n^2 + 46*n + 8)*x^2)*v[i - 1] - ((2*n - 2)*x + (4*n^2 + 4*n - 9)*x^2 + (16*n^2 - 16*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 32*n^2 + 42*n - 18)*x)); 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
• 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Three-term recurrence criterion (above)
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

• A103327 → basis A013611 → θ+1
• A054142 → θ+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