A156366 → Pₙ + Pₙ₊₁

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0: 2,    45,      108,         27;
  1: 2,   111,      693,        729,          81;
  2: 2,   249,     3312,       8856,        4698,          243;
  3: 2,   531,    13437,      73386,      101088,        29403,           729;
  4: 2,  1101,    49356,     486945,     1361610,      1072359,        180792,          2187;
  5: 2,  2247,   170109,    2804031,    13916529,     22484061,      10829295,       1100061,          6561;
  6: 2,  4545,   562032,   14672502,   118790064,    339856884,     342484200,     105723954,       6652854,        19683;
  7: 2,  9147,  1804293,   71784360,   894925908,   4139408286,    7430920074,    4923654336,    1008097650,     40094271,       59049;
  8: 2, 18357,  5678172,  334561671,  6160583628,  43311477762,  125570541600,  149857530894,   67842852642,   9453882681,   241156116,     177147;
  9: 2, 36783, 17621973, 1504470645, 39657600729, 405356511942, 1777071847410, 3437832740634, 2843991552420, 905648704443, 87617437641, 1448885313, 531441;

The row polynomials satisfy P(n,x) = (1+(3*n+3)*x)*P(n-2,x) + (x-3*x^2)*d/dx P(n-2,x) + (3*n+9)*x*P(n-1,x) + (x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 45*x + 108*x^2 + 27*x^3, P(1,x) = 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4.

P[0, x_] := 2 + 45*x + 108*x^2 + 27*x^3;
P[1, x_] := 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n + 3)*x)*P[n - 2, x] + (x - 3*x^2)*D[P[n - 2, x], x] + (3*n + 9)*x*P[n - 1, x] + (x - 3*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] = 2 + 45*x + 108*x^2 + 27*x^3; v[2] = 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n + 3)*x)*v[i - 2] + (x - 3*x^2)*deriv(v[i - 2], x) + (3*n + 9)*x*v[i - 1] + (x - 3*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-4796775b805410d0

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• obreschkoff_combination e-4796775b805410d0

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-e495670ea063ae0d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

  • premise (sequence): Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [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
3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.