A156365 → 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: 2,    30,      48,         8;
  1: 2,    74,     308,       216,         16;
  2: 2,   166,    1472,      2624,        928,          32;
  3: 2,   354,    5972,     21744,      19968,        3872,           64;
  4: 2,   734,   21936,    144280,     268960,      141216,        15872,          128;
  5: 2,  1498,   75604,    830824,    2748944,     2960864,       950720,        64384,          256;
  6: 2,  3030,  249792,   4347408,   23464704,    44754816,     30067200,      6187776,       259584,         512;
  7: 2,  6098,  801908,  21269440,  176775488,   545107264,    652371584,    288169984,     39334400,     1042944,       1024;
  8: 2, 12238, 2523632,  99129384, 1216905408,  5703569088,  11024025600,   8770811136,   2647122432,   245917184,    4182016,     2048;
  9: 2, 24522, 7831988, 445769080, 7833600144, 53380281408, 156011794560, 201208317696, 110968120320, 23558001152, 1519420416, 16750592, 4096;

The row polynomials satisfy P(n,x) = (1+(2*n+2)*x)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + (2*n+6)*x*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 30*x + 48*x^2 + 8*x^3, P(1,x) = 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4.

P[0, x_] := 2 + 30*x + 48*x^2 + 8*x^3;
P[1, x_] := 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n + 2)*x)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + (2*n + 6)*x*P[n - 1, x] + (x - 2*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 + 30*x + 48*x^2 + 8*x^3; v[2] = 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n + 2)*x)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + (2*n + 6)*x*v[i - 1] + (x - 2*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-00ad498f5c132fdf

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-00ad498f5c132fdf

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

  • premise (sequence): T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-4e3e568f497fe445 root: theorem check

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

  • premise (sequence): T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…
    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.