basis-map coefficients of A021012 → basis A008277 → D

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Triangle begins:
   1:          0;
   2:         -1;
   3:          6,           4;
   4:        -42,         -72,          -18;
   5:        360,        1200,          720,           96;
   6:      -3720,      -21600,       -23400,        -7200,         -600;
   7:      45360,      433440,       756000,       403200,        75600,         4320;
   8:    -640080,    -9737280,    -25719120,    -21168000,     -6703200,      -846720,      -35280;
   9:   10281600,   243936000,    939859200,   1121057280,    533433600,    111767040,    10160640,     322560;
  10: -185431680, -6771340800, -37128067200, -61725888000, -41417308800, -12802406400, -1905120000, -130636800, -3265920;

The row polynomials satisfy P(n,x) = -(2*n^2-6*n+4+(2*n^2-6*n+4)*x+(n^2-3*n+2)*x^2)*P(n-2,x) - ((n^2-3*n+2)*x+(n^2-3*n+2)*x^2)*d/dx P(n-2,x) - (3*n-3+(2*n-2)*x)*P(n-1,x) - (n-1)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -1.

P[1, x_] := 0;
P[2, x_] := -1;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] - ((n^2 - 3*n + 2)*x + (n^2 - 3*n + 2)*x^2)*D[P[n - 2, x], x] - (3*n - 3 + (2*n - 2)*x)*P[n - 1, x] - (n - 1)*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] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = -(2*n^2 - 6*n + 4 + (2*n^2 - 6*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] - ((n^2 - 3*n + 2)*x + (n^2 - 3*n + 2)*x^2)*deriv(v[i - 2], x) - (3*n - 3 + (2*n - 2)*x)*v[i - 1] - (n - 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
            • ↑ Parent-row provers of polytool (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
                • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A130123 → basis A008277 → D
• basis-map coefficients of A112227 → basis A008277 → D
• A008277 → D
• A008277 → θ+1 → D+1
• A008277 → θ+1 → D+1 → reverse
• A008277 → θ+1 → D+1

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.