A028338 → x ↦ −2x−1 (shift -1) → Pₙ + Pₙ₊₁

OEIS matches ?

• A033322: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A130713: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A223082: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A355627: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A329679: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A267883: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A236619: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,       1;
   2: 0,       0,      1;
   3: 0,       1,     -2,        1;
   4: 0,      -4,      8,       -5,      1;
   5: 0,      18,    -39,       29,     -9,       1;
   6: 0,     -96,    224,     -190,     75,     -14,     1;
   7: 0,     600,  -1490,     1399,   -650,     160,   -20,     1;
   8: 0,   -4320,  11304,   -11508,   6034,   -1785,   301,   -27,   1;
   9: 0,   35280, -96516,   104992, -60515,   20489, -4214,   518, -35,   1;
  10: 0, -322560, 916992, -1054576, 656396, -246876, 58737, -8904, 834, -44, 1;

The row polynomials satisfy (n-3-x)*P(n,x) = -(n^2-4*n+4-(2*n-4)*x+x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1 + x.

P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 4 - (2*n - 4)*x + x^2)*P[n - 1, x])/(n - 3 - x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (-(n^2 - 4*n + 4 - (2*n - 4)*x + x^2)*v[i - 1])/(n - 3 - 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-fe41916ed6845daa

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-fe41916ed6845daa

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

  • premise (sequence): Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift -1 e-6e459812af5dc78f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-59b849691f316c6f root: theorem check

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

        • parent_provers root_domain e-ef36674c01230a86 root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    • ↵ preserver_image affine, shift -1 e-6e459812af5dc78f (expanded above)

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.