A285061 → 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,        4;
  1: 2,       28,         16;
  2: 2,      148,        256,          64;
  3: 2,      748,       2896,        1856,         256;
  4: 2,     3748,      29056,       35712,       11776,        1024;
  5: 2,    18748,     276496,      580480,      343040,       68608,        4096;
  6: 2,    93748,    2563456,     8652224,     8153600,     2812928,      376832,      16384;
  7: 2,   468748,   23446096,   122732736,   173220096,    91685888,    20672512,    1982464,     65536;
  8: 2,  2343748,  212889856,  1689309952,  3435672576,  2618284032,   883556352,  140181504,  10092544,   262144;
  9: 2, 11718748, 1925383696, 22812588800, 65163673600, 68726654976, 32562044928, 7599489024, 893779968, 50069504, 1048576;

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

P[0, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 1, x] + 4*x*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] = 2 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 1] + 4*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 - 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-f7eb819d34bc450b

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-f7eb819d34bc450b

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

  • premise (sequence): Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-c2ab246647955957 root: theorem check

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

  • premise (sequence): Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).
    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.