basis-map coefficients of A297191, shifted → θ+1 → D+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 0, 6, 3;
  2: 0, 0, 0, 20, 5;
  3: 0, 0, 0,  0, 0, 42, 7;
  4: 0, 0, 0,  0, 0,  0, 0, 72, 9;
  5: 0, 0, 0,  0, 0,  0, 0,  0, 0, 110, 11;
  6: 0, 0, 0,  0, 0,  0, 0,  0, 0,   0,  0, 156, 13;
  7: 0, 0, 0,  0, 0,  0, 0,  0, 0,   0,  0,   0,  0, 210, 15;
  8: 0, 0, 0,  0, 0,  0, 0,  0, 0,   0,  0,   0,  0,   0,  0, 272, 17;
  9: 0, 0, 0,  0, 0,  0, 0,  0, 0,   0,  0,   0,  0,   0,  0,   0,  0, 342, 19;

The row polynomials satisfy (4*n^2-6*n+2+(2*n-1)*x)*P(n,x) = ((4*n^2+2*n)*x^2+(2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 + 2*n)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(4*n^2 - 6*n + 2 + (2*n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (((4*n^2 + 2*n)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(4*n^2 - 6*n + 2 + (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 theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
        • 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 Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• A134082 → reverse
• A134082: Triangle read by rows, (n-1) zeros followed by (2n, 1).
• basis-map coefficients of A297191, shifted → (θ+1)² → D+1
• basis-map coefficients of A297191, shifted → D+1
• A134082 → D

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.