A112007 → 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:        3,         1;
  1:        8,         9,          1;
  2:       30,        66,         23,          1;
  3:      144,       502,        350,         53,          1;
  4:      840,      4152,       4728,       1504,        115,          1;
  5:     5760,     37692,      62540,      33572,       5724,        241,         1;
  6:    45360,    375120,     843444,     676140,     201410,      20190,       495,        1;
  7:   403200,   4075056,   11811600,   13084084,    5961300,    1082450,     67754,     1005,      1;
  8:  3991680,  48072960,  173213208,  251344968,  161122884,   45827500,   5393420,   219852,   2027,    1;
  9: 43545600, 612695520, 2669668128, 4881068856, 4158052472, 1693149304, 319695880, 25462752, 697064, 4073, 1;

The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+1+(n+1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + x, P(1,x) = 8 + 9*x + x^2.

P[0, x_] := 3 + x;
P[1, x_] := 8 + 9*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 1 + (n + 1)*x)*P[n - 1, x] + (x - 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] = 3 + x; v[2] = 8 + 9*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + (n - 1)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 1 + (n + 1)*x)*v[i - 1] + (x - 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-f8a49e57c7ccb041

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-f8a49e57c7ccb041

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

  • premise (sequence): Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-244c1069af538593 root: theorem check

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

  • premise (sequence): Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A163936 → Pₙ + Pₙ₊₁

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.