A256890 → 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:
   1:   12,      64,        56,          8;
   2:   24,     248,       468,        204,          16;
   3:   48,     880,      3060,       2840,         700,          32;
   4:   96,    2960,     17324,      29084,       15364,        2308,          64;
   5:  192,    9616,     89332,     246720,      234960,       76928,        7404,         128;
   6:  384,   30512,    431980,    1844244,     2863640,     1698616,      364668,       23300,         256;
   7:  768,   95248,   1995188,   12602408,    29871268,    28851872,    11335772,     1661032,       72332,        512;
   8: 1536,  293936,   8910060,   80611404,   278422908,   408418620,   262173444,    71270820,     7344276,     222372,      1024;
   9: 3072,  899728,  38801460,  490519760,  2385061640,  5059085696,  4929089984,  2203499600,   428159000,   31751120,    678892,    2048;
  10: 6144, 2738096, 165780716, 2872142660, 19145934480, 56627959632, 79524214728, 54083931288, 17432954640, 2482785120, 134929324, 2062276, 4096;

The row polynomials satisfy P(n,x) = (2+(n+1)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (1+(n+3)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 64*x + 56*x^2 + 8*x^3, P(2,x) = 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4.

P[1, x_] := 12 + 64*x + 56*x^2 + 8*x^3;
P[2, x_] := 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4;
P[n_, x_] := P[n, x] = Expand[(2 + (n + 1)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (1 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 12 + 64*x + 56*x^2 + 8*x^3; v[2] = 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4; for(n = 3, N, my(i = n); v[i] = (2 + (n + 1)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (1 + (n + 3)*x)*v[i - 1] + (x - 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-a7f8a3dfd72049c9

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-a7f8a3dfd72049c9

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

  • premise (sequence): Triangle 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) = x + 2.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-072c1088d2b33d0a root: theorem check

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

  • premise (sequence): Triangle 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) = x + 2.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    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.