A257612 → 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:    3,        2;
  1:    6,       26,           4;
  2:   12,      208,         188,            8;
  3:   24,     1400,        3864,         1224,           16;
  4:   48,     8800,       57504,        55040,         7600,            32;
  5:   96,    53792,      728576,      1560896,       682336,         46240,           64;
  6:  192,   325120,     8438848,     34751616,     33919296,       7810304,       278976,          128;
  7:  384,  1956224,    92656768,    669478272,   1230048128,     643997824,     85171584,      1677440,        256;
  8:  768, 11749888,   984138752,  11750715392,  36736410112,   36175840256,  11199374336,    900923392,   10072832,      512;
  9: 1536, 70527488, 10234174464, 193694607360, 964229047296, 1599282293760, 939803922432, 183479371776, 9345000960, 60455424, 1024;

The row polynomials satisfy P(n,x) = (2+(4*n-6)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (1+(4*n+2)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 26*x + 4*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 6 + 26*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (4*n - 6)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (1 + (4*n + 2)*x)*P[n - 1, x] + (4*x - 4*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] = 3 + 2*x; v[2] = 6 + 26*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (4*n - 6)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (1 + (4*n + 2)*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 - 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-697623ff3ff432a2

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-697623ff3ff432a2

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-1498aa51594830e4 root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.
    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.