A159854 → basis A008277 → (θ+1)²

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Submission text ?

Triangle begins:
  0:   1,     12,       9;
  1:   2,     40,      63,       16;
  2:   4,    128,     342,      192,        25;
  3:   8,    400,    1656,     1568,       450,        36;
  4:  16,   1232,    7524,    10784,      5150,       900,       49;
  5:  32,   3760,   32868,    67296,     47750,     13716,     1617,       64;
  6:  64,  11408,  139932,   394912,    391650,    164772,    31605,     2688,      81;
  7: 128,  34480,  585396,  2223328,   2966950,   1717380,   477113,    65472,    4212,    100;
  8: 256, 103952, 2419164, 12157344,  21275650,  16294068,  6154449,  1212416,  124983,   6300,  121;
  9: 512, 312880, 9910548, 65087456, 146649750, 144695412, 71413629, 18950208, 2784294, 223600, 9075, 144;

The row polynomials satisfy P(n,x) = -(2+8*x-9*x^2)*P(n-2,x) - (4*x-8*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (3+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 12*x + 9*x^2, P(1,x) = 2 + 40*x + 63*x^2 + 16*x^3.

P[0, x_] := 1 + 12*x + 9*x^2;
P[1, x_] := 2 + 40*x + 63*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(2 + 8*x - 9*x^2)*P[n - 2, x] - (4*x - 8*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (3 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*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 + 12*x + 9*x^2; v[2] = 2 + 40*x + 63*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 8*x - 9*x^2)*v[i - 2] - (4*x - 8*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (3 + 4*x)*v[i - 1] + (2*x - 4*x^2)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.