Hadamard-factor parent of A155495 → θ+1

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Triangle begins:
   1: 1,  12,     3;
   2: 1,  30,    45,      4;
   3: 1,  56,   210,    112,       5;
   4: 1,  90,   630,    840,     225,       6;
   5: 1, 132,  1485,   3696,    2475,     396,       7;
   6: 1, 182,  3003,  12012,   15015,    6006,     637,       8;
   7: 1, 240,  5460,  32032,   64350,   48048,   12740,     960,      9;
   8: 1, 306,  9180,  74256,  218790,  262548,  129948,   24480,   1377,    10;
   9: 1, 380, 14535, 155040,  629850, 1108536,  881790,  310080,  43605,  1900,   11;
  10: 1, 462, 21945, 298452, 1598850, 3879876, 4526522, 2558160, 671517, 73150, 2541, 12;

The row polynomials satisfy -(3*n-(2*n^3+3*n^2+n)*x)*P(n,x) = -(3*n-(2*n^3+3*n^2-17*n)*x-(6*n^3+17*n^2+8*n)*x^2)*P(n-1,x) - (6*x-(4*n^2+12*n+12)*x^2+(4*n^2+12*n+6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 12*x + 3*x^2.

P[1, x_] := 1 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - (2*n^3 + 3*n^2 - 17*n)*x - (6*n^3 + 17*n^2 + 8*n)*x^2)*P[n - 1, x] - (6*x - (4*n^2 + 12*n + 12)*x^2 + (4*n^2 + 12*n + 6)*x^3)*D[P[n - 1, x], x])/(-3*n + (2*n^3 + 3*n^2 + n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 12*x + 3*x^2; for(n = 2, N, my(i = n); v[i] = (-(3*n - (2*n^3 + 3*n^2 - 17*n)*x - (6*n^3 + 17*n^2 + 8*n)*x^2)*v[i - 1] - (6*x - (4*n^2 + 12*n + 12)*x^2 + (4*n^2 + 12*n + 6)*x^3)*deriv(v[i - 1], x))/(-3*n + (2*n^3 + 3*n^2 + n)*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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    • 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A034870 → Veronese 2 → θ+1
• A086645 → 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