A019538 → Pₙ + Pₙ₊₁

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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,    2;
   2: 2,    8,      6;
   3: 2,   20,     42,      24;
   4: 2,   44,    186,     264,      120;
   5: 2,   92,    690,    1800,     1920,       720;
   6: 2,  188,   2346,    9960,    18600,     15840,      5040;
   7: 2,  380,   7602,   49224,   142800,    206640,    146160,     40320;
   8: 2,  764,  23946,  227304,   960120,   2096640,   2469600,   1491840,    362880;
   9: 2, 1532,  74130, 1005000,  5937120,  18340560,  31963680,  31691520,  16692480,   3628800;
  10: 2, 3068, 226986, 4316520, 34710600, 145666080, 352129680, 509241600, 435456000, 203212800, 39916800;

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

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

P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; for(n = 2, N, my(i = n); v[i] = (1 + 2*x)*v[i - 1] + (x + x^2)*deriv(v[i - 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-d58e3f8b704ee8a6

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-d58e3f8b704ee8a6

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

  • premise (sequence): Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check ma_wang_interval e-e395982737c903b7 root: theorem check

      Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

    • parent_provers root_domain e-052cfd8551fd4fcd root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-b7f464e4dcf953bf

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

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

        • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): eulerian
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
            • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)
  • premise (sequence): Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A131689 → Pₙ + Pₙ₊₁

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.