A028338 → 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,          1;
  1:         4,          5,          1;
  2:        18,         27,         10,         1;
  3:       120,        199,         95,        17,         1;
  4:      1050,       1865,       1036,       246,        26,        1;
  5:     11340,      21213,      13089,      3710,       530,       37,       1;
  6:    145530,     283731,     189470,     60859,     10550,     1009,      50,      1;
  7:   2162160,    4362447,    3101503,   1095395,    218099,    25613,    1757,     65,    1;
  8:  36486450,   75795345,   56733336,  21608460,   4782988,   651574,   55384,   2860,   82,   1;
  9: 689188500, 1468401525, 1147880385, 465218044, 112069124, 17113614, 1700454, 109596, 4416, 101, 1;

The row polynomials satisfy (2*n+x)*P(n,x) = (4*n^2+2*n-2+(4*n+1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*P[n - 1, x])/(2*n + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*v[i - 1])/(2*n + 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-3733e878156287d8

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-3733e878156287d8

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

  • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-59b849691f316c6f root: theorem check

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

    • parent_provers root_domain e-ef36674c01230a86 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
    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.