A166341 → 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,    33,      24,         1;
   2: 2,    73,     161,        51,         1;
   3: 2,   155,     752,       664,       106,          1;
   4: 2,   321,    2981,      5526,      2472,        217,          1;
   5: 2,   655,   10764,     36395,     33850,       8613,        440,          1;
   6: 2,  1325,   36661,    207797,    346313,     184711,      28703,        887,         1;
   7: 2,  2667,  120144,   1079418,   2946864,    2808348,     931368,      92766,      1782,        1;
   8: 2,  5353,  383549,   5250560,  22113932,   34216442,   20385002,    4439336,    293450,     3573,       1;
   9: 2, 10727, 1202396,  24362301, 151671036,  357464142,  345361632,  136536594,  20306514,   914299,    7156,     1;
  10: 2, 21477, 3721613, 109181495, 973268433, 3338671242, 4888196010, 3144980646, 861094140, 90077377, 2818041, 14323, 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 + 33*x + 24*x^2 + x^3, P(2,x) = 2 + 73*x + 161*x^2 + 51*x^3 + x^4.

P[1, x_] := 2 + 33*x + 24*x^2 + x^3;
P[2, x_] := 2 + 73*x + 161*x^2 + 51*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 + 33*x + 24*x^2 + x^3; v[2] = 2 + 73*x + 161*x^2 + 51*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-26f69690cce5954a

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-26f69690cce5954a

  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-b02efd7745c46534 root: theorem check

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

    • parent_provers root_domain e-2e1314120c3b22a7 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.