A159854 → basis A008277 → D

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Submission text ?

Triangle begins:
  0:     3,       2;
  1:    10,      14,        3;
  2:    32,      76,       36,        4;
  3:   100,     368,      294,       72,        5;
  4:   308,    1672,     2022,      824,      125,       6;
  5:   940,    7304,    12618,     7640,     1905,     198,       7;
  6:  2852,   31096,    74046,    62664,    22885,    3870,     294,      8;
  7:  8620,  130088,   416874,   474712,   238525,   58422,    7161,    416,     9;
  8: 25988,  537592,  2279502,  3404104,  2263065,  753606,  132608,  12344,   567,  10;
  9: 78220, 2202344, 12203898, 23463960, 20096585, 8744526, 2072679, 274992, 20124, 750, 11;

The row polynomials satisfy P(n,x) = -(6+4*x+x^2)*P(n-2,x) - (2*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 10 + 14*x + 3*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 10 + 14*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(6 + 4*x + x^2)*P[n - 2, x] - (2*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 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] = 3 + 2*x; v[2] = 10 + 14*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(6 + 4*x + x^2)*v[i - 2] - (2*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 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 - 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 basis_map_A008277 of Riordan array (1-x,x/(1-x)).?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • 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
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

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