A257614 → 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:    3,         2;
  1:    6,        30,           4;
  2:   12,       272,         248,            8;
  3:   24,      2088,        6100,         1852,            16;
  4:   48,     15104,      107476,       103464,         13276,            32;
  5:   96,    106944,     1612780,      3556700,       1524420,         93716,            64;
  6:  192,    751520,    22174548,     95436960,      93493040,      20757072,        657900,          128;
  7:  384,   5267424,   289674796,   2213585316,    4139184600,    2142266824,     269669244,      4609716,         256;
  8:  768,  36887456,  3667703060,  46797226904,  150368955332,  148443356048,   44944634876,   3403301240,    32278124,       512;
  9: 1536, 258247008, 45538661932, 929912813548, 4795020261052, 8047696393660, 4693374131908, 888635881732, 42172248148, 225969652, 1024;

The row polynomials satisfy P(n,x) = (2+(5*n-8)*x)*P(n-2,x) + (5*x-5*x^2)*d/dx P(n-2,x) + (1+(5*n+2)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 30*x + 4*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 6 + 30*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (5*n - 8)*x)*P[n - 2, x] + (5*x - 5*x^2)*D[P[n - 2, x], x] + (1 + (5*n + 2)*x)*P[n - 1, x] + (5*x - 5*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] = 3 + 2*x; v[2] = 6 + 30*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (5*n - 8)*x)*v[i - 2] + (5*x - 5*x^2)*deriv(v[i - 2], x) + (1 + (5*n + 2)*x)*v[i - 1] + (5*x - 5*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-18b3eb6c762aa18f

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-18b3eb6c762aa18f

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

  • premise (sequence): Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-73c74b4a6ca9fdda root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…
    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.