basis-map coefficients of A030528, shifted → D → D+1

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Submission text ?

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

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

P[0, x_] := 0;
P[1, x_] := -1;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^2 - n)*x + n*x^2)*P[n - 1, x])/(n^2 - 3*n + 2 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-((n^2 - n)*x + n*x^2)*v[i - 1])/(n^2 - 3*n + 2 + (n - 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
        • 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 Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A127648 → D → D+1
• A136572 → θ+1 → D+1
• basis-map coefficients of A026729, shifted → D+1
• A132440 → D+1
• basis-map coefficients of A026729, shifted → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → θ+1 → D+1
• image under θ+1 → θ+1 → D+1 → D
• Hadamard-factor parent of A127773 → D+1
• image under θ+1 → θ+1 → D+1 → reverse
• A131431 → θ+1 → D+1
• A130123 → (θ+1)² → D+1
• basis-map coefficients of A021012 → θ+1 → D+1
• A127648 → (θ+1)² → D+1
• A136572 → D+1
• basis-map coefficients of A026729 → D → D+1
• basis-map coefficients of A026729, shifted → D → D+1
• A130123 → D+1
• basis-map coefficients of A026729, shifted → θ+1 → D+1
• image under θ+1 → θ+1 → D+1
• A127773 → D+1
• image under θ+1 → (θ+1)² → D+1
• image under θ+1 → D+1
• basis-map coefficients of A021012 → D+1
• A131436 → D+1
• A127773 → D → D+1
• basis-map coefficients of A026729 → D+1
• basis-map coefficients of A030528, shifted → (θ+1)² → D+1
• A136572 → (θ+1)² → D+1
• basis-map coefficients of A026729 → (θ+1)² → D+1
• A131431 → D+1
• A127773 → θ+1 → D+1
• A130123 → θ+1 → D+1
• A131431 → D+1 → D
• A131431 → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → D+1
• A130123 → D → D+1
• A023531 → D+1
• A136572 → D → D+1
• A131431 → D+1 → reverse
• A132440 → θ+1 → D+1

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.