A159854 → basis A008277 → D+1

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Submission text ?

Triangle begins:
  0:     4,       5,        1;
  1:    12,      24,       10,        1;
  2:    36,     108,       74,       16,        1;
  3:   108,     468,      478,      170,       23,        1;
  4:   324,    1980,     2858,     1498,      331,       31,       1;
  5:   972,    8244,    16270,    11846,     3815,      579,      40,      1;
  6:  2916,   33948,    89594,    87346,    38551,     8447,     939,     50,     1;
  7:  8748,  138708,   481918,   613670,   357203,   106127,   16898,   1439,    61,    1;
  8: 26244,  563580,  2548298,  4163938,  3114091,  1206219,  258209,  31288,  2110,   73,  1;
  9: 78732, 2280564, 13305070, 27531926, 25962575, 12763843, 3530100, 571089, 54498, 2986, 86, 1;

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

P[0, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 4 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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, Ray basis map theorem
• 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, Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators, Ray basis map theorem
• 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

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
• 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 (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.