A173018 → Pₙ + Pₙ₊₁

OEIS matches ?

• A059171: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A261766: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001048: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:
   1: 2;
   2: 2,    1;
   3: 2,    5,     1;
   4: 2,   15,    12,      1;
   5: 2,   37,    77,     27,       1;
   6: 2,   83,   368,    328,      58,       1;
   7: 2,  177,  1493,   2718,    1248,     121,      1;
   8: 2,  367,  5484,  18035,   16810,    4413,    248,     1;
   9: 2,  749, 18901, 103853,  171809,   92527,  14855,   503,    1;
  10: 2, 1515, 62448, 543426, 1466544, 1398588, 469800, 48342, 1014, 1;

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

P[1, x_] := 2;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n - 1)*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; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n - 1)*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-f5cd00dcd9141f2d

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-f5cd00dcd9141f2d

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

  • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-083b494dcfd2f637 root: theorem check

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

    • parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
    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.

• A008292 → 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. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.