A059364 → x ↦ −x−1 (shift -1) → 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:
   1: 2,   -1;
   2: 2,   -4,    2;
   3: 2,   -9,   13,     -6;
   4: 2,  -16,   46,    -56,     24;
   5: 2,  -25,  120,   -275,    298,     -120;
   6: 2,  -36,  260,   -960,   1898,    -1884,     720;
   7: 2,  -49,  497,  -2695,   8393,   -14896,   13788,    -5040;
   8: 2,  -64,  868,  -6496,  29218,   -80416,  131192,  -114624,    40320;
   9: 2,  -81, 1416, -13986,  85722,  -336609,  841804, -1282284,  1066896,   -362880;
  10: 2, -100, 2190, -27600, 221046, -1171380, 4140610, -9582200, 13780152, -10991520, 3628800;

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

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

P(N) = {my(v = vector(N)); v[1] = 2 - x; for(n = 2, N, my(i = n); v[i] = (-(2 - (3*n - 2)*x + (n^2 - n)*x^2)*v[i - 1])/(-2 + (n - 1)*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-ed573324c4b51ce2

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-ed573324c4b51ce2

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

  • premise (sequence): Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift -1 e-68dd2f09e1ded893

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-50fc3242d79d4e9a root: theorem check

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

        • basis_map A007318 e-b8821d58f365ef74

          Geometric basis map (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-419a6747cb084155 root: theorem check

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

            • preserver_image reversal, shift 0 e-986a227a8a459030

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • theorem_check liu_wang e-5f21311bb67f8c1c root: theorem check

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

                • basis_map A007318 e-0b3ba9047fae59ed

                  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 ?

                • parent_provers root_domain e-2e5c10abc86c264d root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image reversal, shift 0 e-90e69a839bdbf74b

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ theorem_check liu_wang e-419a6747cb084155 (expanded above)
                    • parent_provers paper_theorem e-b29f35705824bef6 root: theorem check

                      Parent-row provers of polytool polytool ?

                • preserver_image affine, shift 0 e-1ba7a8eebd893648

                  Affine, reversal and Möbius substitutions (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 (expanded above)
            • ↵ parent_provers paper_theorem e-b29f35705824bef6 (expanded above)
        • parent_provers root_domain e-bf95690978550a83 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image affine, shift 0 e-8e142d34c9c1ef70

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-419a6747cb084155 (expanded above)
            • ↵ preserver_image reversal, shift 0 e-986a227a8a459030 (expanded above)
            • ↵ parent_provers paper_theorem e-b29f35705824bef6 (expanded above)
  • premise (sequence): Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients …
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    • ↵ preserver_image affine, shift -1 e-68dd2f09e1ded893 (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.