basis-map coefficients of A112227 → basis A008277 → θ+1

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Submission text ?

Triangle begins:
   1: 1,    6,      3;
   2: 1,   14,     18,       4;
   3: 1,   30,     75,      40,       5;
   4: 1,   62,    270,     260,      75,       6;
   5: 1,  126,    903,    1400,     700,     126,       7;
   6: 1,  254,   2898,    6804,    5250,    1596,     196,       8;
   7: 1,  510,   9075,   31080,   34755,   15876,    3234,     288,      9;
   8: 1, 1022,  27990,  136420,  212625,  136962,   41160,    6000,    405,    10;
   9: 1, 2046,  85503,  583000, 1233650, 1076922,  447909,   95040,  10395,   550,  11;
  10: 1, 4094, 259578, 2446004, 6897000, 7941912, 4391772, 1272216, 200475, 17050, 726, 12;

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - x^2*d/dx P(n-2,x) + (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 6*x + 3*x^2, P(2,x) = 1 + 14*x + 18*x^2 + 4*x^3.

P[1, x_] := 1 + 6*x + 3*x^2;
P[2, x_] := 1 + 14*x + 18*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 6*x + 3*x^2; v[2] = 1 + 14*x + 18*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - x^2*deriv(v[i - 2], x) + (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], 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_A008277 of basis-map coefficients of A scaled Hermite triangle.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A008277 → θ+1
• basis-map coefficients of A021012 → basis A008277 → θ+1
• A008277 → θ+1 → reverse
• A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
• A130123 → basis A008277 → θ+1
• A048993 → D+1 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.