A028338 → x ↦ 2x−3 → 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:  0,      1;
  1: -1,      0,      1;
  2:  0,     -2,      1,     1;
  3:  0,     -3,     -1,     3,     1;
  4:  0,     -8,     -6,     7,     6,     1;
  5:  0,    -30,    -31,    20,    30,    10,     1;
  6:  0,   -144,   -180,    64,   165,    79,    15,    1;
  7:  0,   -840,  -1198,   189,  1029,   630,   168,   21,   1;
  8:  0,  -5760,  -9072,    76,  7252,  5369,  1792,  314,  28,  1;
  9:  0, -45360, -77292, -8604, 57203, 49644, 19551, 4284, 537, 36, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 - 2*n + (2*n - 2)*x + x^2)*v[i - 1])/(n - 1 + 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-c8fbb9d7e751998c

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-c8fbb9d7e751998c

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

  • premise (sequence): Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n,…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-0658035b499d17ce

      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 ?

    • basis_map A130595 e-2866478407d5f715

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n,…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-0658035b499d17ce (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.