A144696 → 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,    9,      4;
  2: 2,   25,     37,       8;
  3: 2,   59,    204,     139,       16;
  4: 2,  129,    889,    1339,      489,       32;
  5: 2,  271,   3400,    9630,     7670,     1643,       64;
  6: 2,  557,  12009,   58202,    85292,    40065,     5345,      128;
  7: 2, 1131,  40300,  314189,   767250,   659653,   196064,    16995,     256;
  8: 2, 2281, 130705, 1569829,  5985793,  8483323,  4632259,   914935,   53161,    512;
  9: 2, 4583, 414168, 7424688, 42223188, 92111334, 82704552, 30289992, 4121322, 164347, 1024;

The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 2 + 9*x + 4*x^2.

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

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; v[2] = 2 + 9*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 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 - 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-839c605163c8e6c6

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-839c605163c8e6c6

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

  • premise (sequence): Triangle of 2-Eulerian numbers.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-879aa012f9d1e1ab root: theorem check

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

  • premise (sequence): Triangle of 2-Eulerian numbers.
    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.