basis-map coefficients of A021012 → basis A008277 → D+1

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Triangle begins:
   1:          1;
   2:         -2,          -1;
   3:          8,          10,            2;
   4:        -48,        -114,          -54,           -6;
   5:        384,        1560,         1320,          336,           24;
   6:      -3840,      -25320,       -34200,       -15000,        -2400,         -120;
   7:      46080,      478800,       972720,       655200,       176400,        19440,         720;
   8:    -645120,   -10377360,    -30587760,    -29741040,    -11995200,     -2187360,     -176400,      -5040;
   9:   10321920,   254217600,   1061827200,   1434343680,    813697920,    218453760,    28788480,    1774080,     40320;
  10: -185794560, -6956772480, -40513737600, -74101910400, -56848780800, -21085868160, -4038854400, -402796800, -19595520, -362880;

The row polynomials satisfy P(n,x) = -(2*n-2+(n-1)*x)*P(n-1,x) - (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)*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)*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 ?

• A038338 → basis A013609 → basis A008277
• A356546 → basis A013609 → basis A008277
• A143494 → reverse
• A344565 → basis A008277
• A124860 → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A085880 → basis A013609 → basis A008277
• A003506 → basis A008277
• A356546 → basis A008277
• A038330 → basis A013611 → basis A008277
• A196347 → basis A013609 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A038325 → basis A013609 → basis A008277
• A038325 → basis A008277
• A134058 → basis A008277
• A178820 → basis A008277
• A120910 → basis A013610 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A376827 → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A135089 → basis A008277
• A038221 → basis A008277
• A028326 → D → basis A008277
• A196347 → basis A008277
• A028326 → basis A013609 → basis A008277
• A028326 → basis A008277
• A132200 → basis A008277
• A038221 → basis A013609 → basis A008277
• A038208 → D → basis A008277
• A085881 → basis A008277
• A178821 → basis A013609 → basis A008277
• A135065 → basis A008277
• A038338 → basis A008277
• A178822 → basis A008277
• A094305 → basis A008277
• A156992 → basis A008277 → reverse
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A156992 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A085881 → basis A013609 → basis A008277
• A085880 → D → basis A008277
• A258758 → basis A008277
• A038208 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A003506 → D → basis A008277
• A134059 → basis A008277
• A126351 → reverse
• A135838 → basis A008277
• A134346 → basis A008277
• A178821 → basis A008277
• A178820 → basis A013609 → basis A008277
• A156992 → basis A013609 → basis A008277
• A178822 → basis A013609 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A085880 → basis A008277
• A027465 → basis A013611 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277

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.