A145324 → Pₙ + Pₙ₊₁

OEIS matches ?

• A171005: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 3 ?

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:
  0: 2,   2;
  1: 2,   7,    6;
  2: 2,  14,   32,    24;
  3: 2,  23,   97,   178,    120;
  4: 2,  34,  226,   734,   1164,     720;
  5: 2,  47,  450,  2245,   6148,    8748,     5040;
  6: 2,  62,  806,  5690,  23528,   56888,    74304,    40320;
  7: 2,  79, 1337, 12649,  73073,  263536,   578268,   704016,   362880;
  8: 2,  98, 2092, 25508, 195538,  975842,  3164768,  6417432,  7362720,  3628800;
  9: 2, 119, 3126, 47694, 467502, 3072111, 13684354, 40720876, 77290776, 84255840, 39916800;

The row polynomials satisfy (2+(n+1)*x)*P(n,x) = (2+(3*n+4)*x+(n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 2*x.

P[0, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (3*n + 4)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(2 + (n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (3*n + 4)*x + (n^2 + 3*n + 2)*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 - 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-318fe044440ebc67

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-318fe044440ebc67

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

  • premise (sequence): Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-4a8bd3a5d986eb7b root: theorem check

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

    • parent_provers paper_theorem e-4000c0d63fa750fb root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-07dfe744e3acc2e8

      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 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)
  • premise (sequence): Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
    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.