A225117 → 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:    3,        1;
   2:    6,       14,          1;
   3:   12,      106,         61,           1;
   4:   24,      638,       1191,         252,           1;
   5:   48,     3478,      15629,       10929,        1019,           1;
   6:   96,    18110,     168611,      273116,       90016,        4090,           1;
   7:  192,    92278,    1629849,     5152725,     3959350,      696588,       16377,          1;
   8:  384,   465422,   14746543,    82295220,   121629011,    51168502,     5191389,      65528,         1;
   9:  768,  2336326,  127980325,  1181004529,  3004858813,  2439430621,   613435303,   37797739,    262135,       1;
  10: 1536, 11702366, 1081085787, 15766804548, 64274894634, 89194785588, 43829876124, 6988175988, 271203102, 1048566, 1;

The row polynomials satisfy P(n,x) = (2+(3*n-8)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (1+(3*n-2)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + x, P(2,x) = 6 + 14*x + x^2.

P[1, x_] := 3 + x;
P[2, x_] := 6 + 14*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (3*n - 8)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (1 + (3*n - 2)*x)*P[n - 1, x] + (3*x - 3*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] = 3 + x; v[2] = 6 + 14*x + x^2; for(n = 3, N, my(i = n); v[i] = (2 + (3*n - 8)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (1 + (3*n - 2)*x)*v[i - 1] + (3*x - 3*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-998f3ee2c5950862

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-998f3ee2c5950862

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

  • premise (sequence): Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-5f6d53acefa22cfd root: theorem check

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

  • premise (sequence): Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.
    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.