A257621 → 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,        45,           9;
  2:    36,       435,         402,           27;
  3:   108,      3549,        9039,         3183,            81;
  4:   324,     26787,      151092,       145602,         23712,           243;
  5:   972,    194637,     2158341,      4415826,       2039526,        171249,           729;
  6:  2916,   1387731,    28123878,    106676085,     104418600,      26278989,       1217454,         2187;
  7:  8748,   9801597,   346147371,   2238926889,    3993705495,    2162632167,     321256113,      8587059,        6561;
  8: 26244,  68908611,  4106603688,  42825450420,  126966112032,  125211333426,   41008965624,   3795149172,    60330300,     19683;
  9: 78732, 483357549, 47550102465, 768406313076, 3560097193212, 5784516034374, 3477711310206, 731503951140, 43823861904, 423053493, 59049;

The row polynomials satisfy P(n,x) = (3+(4*n-5)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (2+(4*n+3)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 45*x + 9*x^2.

P[0, x_] := 4 + 3*x;
P[1, x_] := 12 + 45*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + (4*n - 5)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (2 + (4*n + 3)*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] = 4 + 3*x; v[2] = 12 + 45*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (4*n - 5)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (2 + (4*n + 3)*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-11ecc0603e43ec86

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-11ecc0603e43ec86

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

  • premise (sequence): Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-53870c9c687f3bf2 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(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…
    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.