A290316 → 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,       6;
   2: 2,      54,         90;
   3: 2,     282,       2304,          2160;
   4: 2,    1230,      29646,        116748,          71280;
   5: 2,    5058,     288180,       2906388,        6841908,          2993760;
   6: 2,   20406,    2439378,      51048144,      287422992,        457131600,        152681760;
   7: 2,   81834,   19145160,     739968264,     8127435024,      29611227672,      34389146160,        9160905600;
   8: 2,  327582,  143787510,    9543015540,   182484708840,    1248961662504,    3221396103384,     2881145052000,      632102486400;
   9: 2, 1310610, 1051522812,  114104709180,  3533641311780,   40645401319944,  191822672910240,   372022960709664,   266297285383200,    49303993939200;
  10: 2, 5242758, 7564329018, 1296492735960, 61866777582360, 1113623591602824, 8622058665339576, 29994920973878256, 45668147294459520, 26933673382329600, 4289447472710400;

The row polynomials satisfy P(n,x) = (1+(18*n-48)*x)*P(n-2,x) + (3*x-9*x^2)*d/dx P(n-2,x) + (18*n-12)*x*P(n-1,x) + (3*x-9*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 6*x, P(2,x) = 2 + 54*x + 90*x^2.

P[1, x_] := 2 + 6*x;
P[2, x_] := 2 + 54*x + 90*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (18*n - 48)*x)*P[n - 2, x] + (3*x - 9*x^2)*D[P[n - 2, x], x] + (18*n - 12)*x*P[n - 1, x] + (3*x - 9*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 + 6*x; v[2] = 2 + 54*x + 90*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (18*n - 48)*x)*v[i - 2] + (3*x - 9*x^2)*deriv(v[i - 2], x) + (18*n - 12)*x*v[i - 1] + (3*x - 9*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-0f78bcf93daf4e45

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-0f78bcf93daf4e45

  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-d08b8ec753afe0a3 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.