basis-map coefficients of A112227 → D → D+1

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Submission text ?

Triangle begins:
   1: 2, 2;
   2: 0, 6,  3;
   3: 0, 0, 12,  4;
   4: 0, 0,  0, 20,  5;
   5: 0, 0,  0,  0, 30,  6;
   6: 0, 0,  0,  0,  0, 42,  7;
   7: 0, 0,  0,  0,  0,  0, 56,  8;
   8: 0, 0,  0,  0,  0,  0,  0, 72,  9;
   9: 0, 0,  0,  0,  0,  0,  0,  0, 90,  10;
  10: 0, 0,  0,  0,  0,  0,  0,  0,  0, 110, 11;

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

P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 + n)*x + (n + 1)*x^2)*P[n - 1, x])/(n^2 - n + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; for(n = 2, N, my(i = n); v[i] = (((n^2 + n)*x + (n + 1)*x^2)*v[i - 1])/(n^2 - n + n*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 derivative of basis-map coefficients of A scaled Hermite triangle.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? 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)

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