A257617 → 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,         38,            4;
  2:   12,        424,          392,              8;
  3:   24,       4064,        12804,           3684,             16;
  4:   48,      37240,       295620,         286880,          33580,              32;
  5:   96,     336824,      5821388,       13310588,        5571748,          303292,             64;
  6:  192,    3035416,    105369748,      478802736,      471313536,       100134824,        2732220,            128;
  7:  384,   27328088,   1817351452,    14872376316,    28412296472,     14506432528,     1720252044,       24596060,          256;
  8:  768,  245974168,  30452230628,   421329298208,  1399386390388,   1385846470232,   408640217132,    28759306672,    221378492,        512;
  9: 1536, 2213815640, 501330020268, 11231614981596, 60419234626044, 102857990195100, 59454717454020, 10853426995092, 472816687764, 1992437916, 1024;

The row polynomials satisfy P(n,x) = (2+(7*n-12)*x)*P(n-2,x) + (7*x-7*x^2)*d/dx P(n-2,x) + (1+(7*n+2)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 38*x + 4*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 6 + 38*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (7*n - 12)*x)*P[n - 2, x] + (7*x - 7*x^2)*D[P[n - 2, x], x] + (1 + (7*n + 2)*x)*P[n - 1, x] + (7*x - 7*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 + 38*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (7*n - 12)*x)*v[i - 2] + (7*x - 7*x^2)*deriv(v[i - 2], x) + (1 + (7*n + 2)*x)*v[i - 1] + (7*x - 7*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-b3016eed90e91248

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-b3016eed90e91248

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-f4ea85fa593cfea8 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.
    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.