A196347 → 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:       3,        5,         2;
  2:       8,       22,        20,         6;
  3:      30,      114,       162,       102,        24;
  4:     144,      696,      1344,      1296,       624,       120;
  5:     840,     4920,     12000,     15600,     11400,      4440,       720;
  6:    5760,    39600,    116640,    190800,    187200,    110160,     36000,      5040;
  7:   45360,   357840,   1234800,   2434320,   2998800,   2363760,   1164240,    327600,     40320;
  8:  403200,  3588480,  14192640,  32739840,  48545280,  47980800,  31610880,  13386240,   3306240,   362880;
  9: 3991680, 39553920, 176359680, 465937920, 807770880, 960180480, 792529920, 448519680, 166561920, 36650880, 3628800;

The row polynomials satisfy (n+1+n*x)*P(n,x) = (n^2+2*n+(2*n^2+3*n)*x+(n^2+n)*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[((n^2 + 2*n + (2*n^2 + 3*n)*x + (n^2 + n)*x^2)*P[n - 1, x])/(n + 1 + n*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] = ((n^2 + 2*n + (2*n^2 + 3*n)*x + (n^2 + n)*x^2)*v[i - 1])/(n + 1 + 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-34b0075c2d0a2285

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-34b0075c2d0a2285

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

  • premise (sequence): Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-61ffea65f7eec286 root: theorem check

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

    • basis_map A007318 e-83332bc01455e815

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers root_domain e-8128a83cc2062067 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-7d7235cf57d80f69

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
    • parent_provers root_domain e-bf1e7898c429e3eb root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-ab4aac9d5a3bd564

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
        • ↵ preserver_image mobius, shift 0 e-7d7235cf57d80f69 (expanded above)
  • premise (sequence): Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, 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.