A225470 → 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:           3,           1;
  1:          12,           8,           1;
  2:          90,          73,          16,           1;
  3:         960,         872,         246,          27,          1;
  4:       13200,       12970,        4271,         621,         41,         1;
  5:      221760,      231272,       84884,       14750,       1315,        58,       1;
  6:     4398240,     4810708,     1916832,      378099,      40930,      2472,      78,      1;
  7:   100531200,   114387200,    48655312,    10570644,    1315664,     97615,    4263,    101,     1;
  8:  2601244800,  3060823600,  1373917348,   322400224,   44658943,   3846409,  208222,   6886,   127,   1;
  9: 75147072000, 91035742400, 42763968720, 10692903744, 1613681544, 155919204, 9872289, 407616, 10566, 156, 1;

The row polynomials satisfy (3*n+x)*P(n,x) = (9*n^2+6*n-3+(6*n+2)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + x.

P[0, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 6*n - 3 + (6*n + 2)*x + x^2)*P[n - 1, x])/(3*n + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 + 6*n - 3 + (6*n + 2)*x + x^2)*v[i - 1])/(3*n + 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-763e8ee49ac3a70c

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-763e8ee49ac3a70c

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

  • premise (sequence): Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-9b3c7c87e4baebfb root: theorem check

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

    • basis_map A007318 e-8562aab4e97e2e3f

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-1844ef33b5ef77b8 root: theorem check

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

        • parent_provers root_domain e-576aa1dc371fa216 root: theorem check

          Parent-row provers of polytool polytool ?

        • basis_map A007318 e-fe032cf3f375eb62

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers root_domain e-aa6a1f197b91b3b6 root: theorem check

              Parent-row provers of polytool polytool ?

    • parent_provers root_domain e-c0733b4d3f559bd0 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
    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.