A290319 → 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:            6,            7,            1;
  2:           50,           65,           16,           1;
  3:          630,          871,          269,          29,          1;
  4:        10530,        15201,         5384,         758,         46,         1;
  5:       218790,       326503,       127249,       21190,       1720,        67,        1;
  6:      5429970,      8323809,      3487208,      654079,      64010,      3391,       92,      1;
  7:    156633750,    245571975,    108964365,    22373659,    2503669,    162085,     6055,    121,     1;
  8:   5148029250,   8228462145,   3827935664,   844760636,  104745396,   7839174,   361536,  10044,   154,   1;
  9: 189871431750, 308650887495, 149439684609, 34997119036, 4710574952, 394159794, 21192150, 732684, 15738, 191, 1;

The row polynomials satisfy (4*n-2+x)*P(n,x) = (16*n^2-4*n-6+(8*n-1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 4*n - 6 + (8*n - 1)*x + x^2)*P[n - 1, x])/(4*n - 2 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 - 4*n - 6 + (8*n - 1)*x + x^2)*v[i - 1])/(4*n - 2 + 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-0f7f0ded98411894

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-0f7f0ded98411894

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

  • premise (sequence): Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-d713510abde08ecc root: theorem check

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

    • parent_provers root_domain e-0a4154652e7b3f7f root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…
    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.