A059364 → 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:        3,         1;
  1:        8,         8,          2;
  2:       30,        53,         31,          6;
  3:      144,       372,        358,        152,         24;
  4:      840,      2882,       3933,       2667,        898,        120;
  5:     5760,     24768,      44168,      41792,      22118,       6204,        720;
  6:    45360,    235188,     520560,     637387,     466093,     203464,      49068,      5040;
  7:   403200,   2452320,    6503768,    9821128,    9233418,    5532592,    2062520,    437184,     40320;
  8:  3991680,  27898704,   86415588,  155667080,  179681367,  137784453,   70172800,  22881132,   4332816,   362880;
  9: 43545600, 344200320, 1221295392, 2561246976, 3515154256, 3298318872, 2142395946, 950974136, 275999832, 47279520, 3628800;

The row polynomials satisfy (n+2+n*x)*P(n,x) = (n^2+4*n+3+(2*n^2+5*n+1)*x+(n^2+n)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + x.

P[0, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (2*n^2 + 5*n + 1)*x + (n^2 + n)*x^2)*P[n - 1, x])/(n + 2 + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n^2 + 5*n + 1)*x + (n^2 + n)*x^2)*v[i - 1])/(n + 2 + n*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-bcc91de7315c2529

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-bcc91de7315c2529

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

  • 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, +inf)
    • 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 T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-8e142d34c9c1ef70 (expanded above)
    • ↵ basis_map A007318 e-b8821d58f365ef74 (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.