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

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Submission text ?

Triangle begins:
   1:       1;
   2:      -1,         -2;
   3:       2,         12,            6;
   4:      -6,        -84,         -108,          -24;
   5:      24,        720,         1800,          960,          120;
   6:    -120,      -7440,       -32400,       -31200,        -9000,         -720;
   7:     720,      90720,       650160,      1008000,       504000,        90720,         5040;
   8:   -5040,   -1280160,    -14605920,    -34292160,    -26460000,     -8043840,      -987840,      -40320;
   9:   40320,   20563200,    365904000,   1253145600,   1401321600,    640120320,    130394880,    11612160,     362880;
  10: -362880, -370863360, -10157011200, -49504089600, -77157360000, -49700770560, -14936140800, -2177280000, -146966400, -3628800;

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

P[1, x_] := 1;
P[2, x_] := -1 - 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 3*n + 2)*x^2*P[n - 2, x] - (n^2 - 3*n + 2)*x^2*D[P[n - 2, x], x] - (n - 1 + (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] = 1; v[2] = -1 - 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 3*n + 2)*x^2*v[i - 2] - (n^2 - 3*n + 2)*x^2*deriv(v[i - 2], x) - (n - 1 + (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 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

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 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 ?

• 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).
• basis-map coefficients of A112227 → basis A008277 → θ+1
• A048993 → D+1 → θ+1
• A130123 → basis A008277 → θ+1
• A008277 → 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. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.