A039756 → reverse → D+1

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Submission text ?

Triangle begins:
  0:    1;
  1:    2,      1;
  2:    5,      6,       1;
  3:   14,     31,      12,       1;
  4:   41,    156,     106,      20,       1;
  5:  122,    781,     840,     270,      30,      1;
  6:  365,   3906,    6331,    3100,     575,     42,     1;
  7: 1094,  19531,   46452,   32711,    9030,   1085,    56,    1;
  8: 3281,  97656,  335476,  328440,  126966,  22344,  1876,   72,  1;
  9: 9842, 488281, 2398800, 3195340, 1666980, 403662, 49056, 3036, 90, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x^2)*v[i - 2] - (2*x - 2*x^2)*deriv(v[i - 2], x) + 4*v[i - 1] + 2*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 - 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from reversal of Triangle of B-analogs of Stirling numbers of 2nd kind.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of B-analogs of Stirling numbers of 2nd kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 reversal of Triangle of B-analogs of Stirling numbers of 2nd kind.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of B-analogs of Stirling numbers of 2nd kind.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A039755 → 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.