A142458 → 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,      1;
  1: 2,      9,         1;
  2: 2,     47,        40,          1;
  3: 2,    205,       585,        167,           1;
  4: 2,    843,      6028,       5648,         678,           1;
  5: 2,   3401,     52657,     115122,       47852,        2725,           1;
  6: 2,  13639,    420984,    1819315,     1756850,      376533,       10916,          1;
  7: 2,  54597,   3197857,   24787369,    46161225,    23451503,     2834963,      43683,         1;
  8: 2, 218435,  23553388,  307511546,   991564804,   970190948,   287257900,   20773022,    174754,      1;
  9: 2, 873793, 170203545, 3584817852, 18663828744, 31254737382, 18001149342, 3321113340, 149648958, 699041, 1;

The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (3*n+1)*x*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 9*x + x^2.

P[0, x_] := 2 + x;
P[1, x_] := 2 + 9*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (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, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 2 + 9*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n - 5)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (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 - 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-3646695af547e3ef

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-3646695af547e3ef

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

  • premise (sequence): Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-98e347ce1f066bf3 root: theorem check

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

  • premise (sequence): Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).
    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.