A257627 → 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:     4,          3;
  1:    12,         63,             9;
  2:    36,        813,           762,             27;
  3:   108,       8931,         26355,           8205,              81;
  4:   324,      92469,        657108,         639684,           84372,             243;
  5:   972,     936435,      13980129,       30943794,        13432914,          852387,             729;
  6:  2916,    9406389,     272656686,     1171614513,      1154650848,       260160207,         8554974,           2187;
  7:  8748,   94210419,    5052486159,    38386262019,     71902040175,     37504267857,      4801732341,       85658361,          6561;
  8: 26244,  942605013,   90731326776,  1146817009170,   3679652086572,   3646136308416,   1114365227472,    86023804854,     856955400,      19683;
  9: 78732, 9427736307, 1597446657741, 32208084965760, 165415265377824, 277884633843894, 162915946078578, 31184451065064, 1512365457900, 8570807151, 59049;

The row polynomials satisfy P(n,x) = (3+(7*n-11)*x)*P(n-2,x) + (7*x-7*x^2)*d/dx P(n-2,x) + (2+(7*n+3)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 63*x + 9*x^2.

P[0, x_] := 4 + 3*x;
P[1, x_] := 12 + 63*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + (7*n - 11)*x)*P[n - 2, x] + (7*x - 7*x^2)*D[P[n - 2, x], x] + (2 + (7*n + 3)*x)*P[n - 1, x] + (7*x - 7*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] = 4 + 3*x; v[2] = 12 + 63*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (7*n - 11)*x)*v[i - 2] + (7*x - 7*x^2)*deriv(v[i - 2], x) + (2 + (7*n + 3)*x)*v[i - 1] + (7*x - 7*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-68ca759bf35f8a33

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-68ca759bf35f8a33

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

  • premise (sequence): Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-4d5a49b3ac34613a 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) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.
    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.