A290315 → 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: 2,      2;
   2: 2,     18,        12;
   3: 2,     82,       296,         120;
   4: 2,    290,      3156,        6072,         1680;
   5: 2,    930,     23952,      122288,       147744,         30240;
   6: 2,   2866,    153308,     1647408,      4950720,       4160544,         665280;
   7: 2,   8690,    892728,    17802152,    106227296,     212921952,      133223040,       17297280;
   8: 2,  26178,   4915620,   168193640,   1762696080,    6765250656,     9789891264,     4783720320,      518918400;
   9: 2,  78658,  26148896,  1456377440,  24816552640,  161741180096,   436799783424,   481870775808,   190464606720,   17643225600;
  10: 2, 236114, 136093356, 11887228256, 312428994560, 3205222846656, 14359969237632, 28990792140288, 25365822617088, 8330583421440, 670442572800;

The row polynomials satisfy P(n,x) = (1+(8*n-22)*x)*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) + (8*n-6)*x*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 2 + 18*x + 12*x^2.

P[1, x_] := 2 + 2*x;
P[2, x_] := 2 + 18*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 22)*x)*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, x], x] + (8*n - 6)*x*P[n - 1, x] + (2*x - 4*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] = 2 + 2*x; v[2] = 2 + 18*x + 12*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (8*n - 22)*x)*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 2], x) + (8*n - 6)*x*v[i - 1] + (2*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, 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-8644a9f1d4c74332

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-8644a9f1d4c74332

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

  • premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-f3090409c727b3ad root: theorem check

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

  • premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
    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.