A049029 → 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:             6,             1;
   2:            50,            16,             1;
   3:           630,           270,            31,            1;
   4:         10530,          5430,           855,           51,           1;
   5:        218790,        128970,         24975,         2075,          76,          1;
   6:       5429970,       3551310,        799995,        82875,        4275,        106,        1;
   7:     156633750,     111474090,      28254555,      3443895,      223650,       7875,      141,       1;
   8:    5148029250,    3933042750,    1097280135,    151911375,    11702145,     522480,    13370,     181,     1;
   9:  189871431750,  154171467450,   46617053175,   7160282955,   630821520,   33615225,  1096830,   21330,   226,   1;
  10: 7764742235250, 6649773079950, 2154425548275, 361075309875, 35501320575, 2175063030, 85113315, 2119950, 32400, 276, 1;

The row polynomials satisfy P(n,x) = (4*n-7+x)*P(n-2,x) + x*d/dx P(n-2,x) + (4*n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 6 + x, P(2,x) = 50 + 16*x + x^2.

P[1, x_] := 6 + x;
P[2, x_] := 50 + 16*x + x^2;
P[n_, x_] := P[n, x] = Expand[(4*n - 7 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (4*n + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 6 + x; v[2] = 50 + 16*x + x^2; for(n = 3, N, my(i = n); v[i] = (4*n - 7 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (4*n + x)*v[i - 1] + x*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-80d7320f7911dda9

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-80d7320f7911dda9

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

  • premise (sequence): Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-2c188005bc3e1dfc root: theorem check

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

  • premise (sequence): Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
    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.