A166340 → 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:
   1: 2,    27,      20,        1;
   2: 2,    61,     133,       43,         1;
   3: 2,   131,     624,      552,        90,          1;
   4: 2,   273,    2485,     4590,      2064,        185,          1;
   5: 2,   559,    9004,    30275,     28170,       7213,        376,          1;
   6: 2,  1133,   30741,   173149,    288145,     153983,      24087,        759,         1;
   7: 2,  2283,  100912,   900754,   2453424,    2338428,     777512,      77958,      1526,        1;
   8: 2,  4585,  322525,  4386600,  18425444,   28489162,   16987770,    3710000,    246850,     3061,       1;
   9: 2,  9191, 1011900, 20371925, 126466220,  297721606,  287657888,  113865050,  16984450,   769635,    6132,     1;
  10: 2, 18405, 3133741, 91361375, 812045625, 2781825426, 4071358770, 2620633758, 718552500, 75388825, 2373297, 12275, 1;

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 >= 3, with P(1,x) = 2 + 27*x + 20*x^2 + x^3, P(2,x) = 2 + 61*x + 133*x^2 + 43*x^3 + x^4.

P[1, x_] := 2 + 27*x + 20*x^2 + x^3;
P[2, x_] := 2 + 61*x + 133*x^2 + 43*x^3 + x^4;
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, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2 + 27*x + 20*x^2 + x^3; v[2] = 2 + 61*x + 133*x^2 + 43*x^3 + x^4; for(n = 3, N, my(i = n); 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, 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-8df0a750c7bea2d2

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-8df0a750c7bea2d2

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

  • premise (sequence): Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-7ffc7a4c532b7fe3 root: theorem check

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

    • parent_provers root_domain e-9eddb58049332ffc root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    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.