A290595 → 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:           2,            2;
   2:           5,           21,             4;
   3:          32,          241,           151,              8;
   4:         308,         3416,          4275,            895,             16;
   5:        3920,        58218,        117694,          53425,           4851,            32;
   6:       61880,      1162360,       3382354,        2615214,         560519,         25199,            64;
   7:     1164800,     26633528,     103813200,      120080636,       45971324,       5326317,        128267,          128;
   8:    25450880,    689131872,    3426311788,     5469829724,     3234798840,     701015740,      47665523,       646647,        256;
   9:   632952320,  19877517040,  121687486672,   253846259260,   212185861588,   73175662990,    9744502570,    410684953,    3245395,      512;
  10: 17649632000, 632494795520, 4642221870720, 12165089548656, 13538734563828, 6742057862196, 1470476644350, 127168539450, 3452299839, 16254367, 1024;

The row polynomials satisfy P(n,x) = (3*n-8+(3*n-7)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (3*n-3+(3*n-1)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 5 + 21*x + 4*x^2.

P[1, x_] := 2 + 2*x;
P[2, x_] := 5 + 21*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(3*n - 8 + (3*n - 7)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (3*n - 3 + (3*n - 1)*x)*P[n - 1, x] + (3*x - 3*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] = 2 + 2*x; v[2] = 5 + 21*x + 4*x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 8 + (3*n - 7)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (3*n - 3 + (3*n - 1)*x)*v[i - 1] + (3*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, 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-35685a58b6434e5d

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-35685a58b6434e5d

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

  • premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-8066ebfb4deadcc4 root: theorem check

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

  • premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
    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.