A120910 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168729: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A027327: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168777: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A052156: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168681: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168825: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A003946: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A025579: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    9,     3;
  1:   18,    15,     3;
  2:   36,    48,    21,     3;
  3:   72,   132,    90,    27,     3;
  4:  144,   336,   312,   144,    33,     3;
  5:  288,   816,   960,   600,   210,    39,     3;
  6:  576,  1920,  2736,  2160,  1020,   288,    45,    3;
  7: 1152,  4416,  7392,  7056,  4200,  1596,   378,   51,   3;
  8: 2304,  9984, 19200, 21504, 15456,  7392,  2352,  480,  57,  3;
  9: 4608, 22272, 48384, 62208, 52416, 30240, 12096, 3312, 594, 63, 3;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1]); 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-a8c0a3645139a0f5

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-a8c0a3645139a0f5

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-26e15e96a906c1df root: theorem check

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

    • parent_provers periodic_product e-3e9b67ef5f31843c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-b63d4680864933c7

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-f5018d5f4ba5a310 root: theorem check

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

        • preserver_image reversal, shift 0 e-98e1992c4c48b815

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-26e15e96a906c1df (expanded above)
            • ↵ parent_provers periodic_product e-3e9b67ef5f31843c (expanded above)
        • parent_provers periodic_product e-f91a02f39edc297f root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A038207 → Pₙ + Pₙ₊₁

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.