basis-map coefficients of A021012 → basis A008277 → x ↦ x+1

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Submission text ?

Triangle begins:
   1:            1;
   2:           -2,            -1;
   3:           10,            10,             2;
   4:          -90,          -132,           -54,            -6;
   5:         1248,          2376,          1464,           336,           24;
   6:       -24360,        -56520,        -46200,        -16200,        -2400,         -120;
   7:       631440,       1717920,       1739520,        820800,       187200,        19440,         720;
   8:    -20865600,     -64849680,     -77968800,     -46146240,    -14288400,     -2293200,     -176400,      -5040;
   9:    852647040,    2970817920,    4121308800,    2926828800,   1146741120,    251193600,    29917440,    1774080,     40320;
  10: -42085008000, -162069465600, -254128855680, -210081755520, -99569191680, -27761408640, -4526565120, -415860480, -19595520, -362880;

The row polynomials satisfy P(n,x) = -(2*n-2+(n-1)*x)*P(n-1,x) - (n-1+(n-1)*x)*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[-(2*n - 2 + (n - 1)*x)*P[n - 1, x] - (n - 1 + (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] = 1; for(n = 2, N, my(i = n); v[i] = -(2*n - 2 + (n - 1)*x)*v[i - 1] - (n - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) 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 ?

• basis-map coefficients of A112227 → basis A008277 → x ↦ x+1
• A008278 → x ↦ x/(x+1)
• A137597 → reverse
• A137597: Triangle read by rows: A008277 * A007318.
• A008277 → x ↦ x+1
• A130123 → basis A008277 → x ↦ x+1
• A008277 → x ↦ x+1 → reverse

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