A142459 → 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,       1;
   2: 2,      11,          1;
   3: 2,      69,         60,           1;
   4: 2,     367,       1121,         309,            1;
   5: 2,    1865,      14624,       13870,         1558,            1;
   6: 2,    9363,     162089,      366174,       150084,         7807,            1;
   7: 2,   46861,    1649196,     7461527,      7257442,      1508475,        39056,           1;
   8: 2,  234359,   15983065,   131038859,    248812025,    125430613,     14521451,      195305,          1;
   9: 2, 1171857,  150487776,  2097079116,   6953184804,   6835411638,   1987631568,   136200684,     976554,       1;
  10: 2, 5859355, 1392280049, 31568244428, 170135399792, 288593712842, 165397067270, 29731100636, 1257251222, 4882803, 1;

The row polynomials satisfy P(n,x) = (1+(4*n-11)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (4*n-3)*x*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 2 + 11*x + x^2.

P[1, x_] := 2 + x;
P[2, x_] := 2 + 11*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 11)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (4*n - 3)*x*P[n - 1, x] + (4*x - 4*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 + x; v[2] = 2 + 11*x + x^2; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 11)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (4*n - 3)*x*v[i - 1] + (4*x - 4*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-2762452c13f28e99

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-2762452c13f28e99

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

  • premise (sequence): Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-ce13bfb75f31b51b root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
    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.