A078812 → reverse → θ+1

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Submission text ?

Triangle begins:
   1: 1;
   2: 1,  4;
   3: 1,  8,   9;
   4: 1, 12,  30,   16;
   5: 1, 16,  63,   80,   25;
   6: 1, 20, 108,  224,  175,    36;
   7: 1, 24, 165,  480,  630,   336,    49;
   8: 1, 28, 234,  880, 1650,  1512,   588,   64;
   9: 1, 32, 315, 1456, 3575,  4752,  3234,  960,   81;
  10: 1, 36, 408, 2240, 6825, 12012, 12012, 6336, 1485, 100;

The row polynomials satisfy (4*n^2-6*n+2+(8*n^3-20*n^2+16*n-4)*x)*P(n,x) = (4*n^2-6*n+2+(8*n^3+4*n^2-28*n+12)*x+(24*n^3-36*n^2-8*n+16)*x^2)*P(n-1,x) - ((2*n-1)*x+(4*n^2+8*n-6)*x^2+(16*n^2-8)*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[((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*P[n - 1, x] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*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] = ((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*v[i - 1] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*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

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-1, 2*k-1).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A091044 → basis A013611 → θ+1
• A172431 → 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