A322944 → 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,         1;
   2:       3,         7,          1;
   3:       8,        44,         16,          1;
   4:      30,       310,        203,         29,         1;
   5:     144,      2472,       2528,        608,        46,         1;
   6:     840,     22120,      32620,      11620,      1435,        67,        1;
   7:    5760,    219840,     443840,     217960,     38840,      2908,       92,      1;
   8:   45360,   2404080,    6403320,    4138120,    993370,    105658,     5299,    121,     1;
   9:  403200,  28694400,   98040320,   80682560,  24961440,   3526320,   248640,   8928,   154,   1;
  10: 3991680, 371226240, 1591349760, 1627077760, 628878320, 113016624, 10483032, 525288, 14163, 191, 1;

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

P[1, x_] := 2 + x;
P[2, x_] := 3 + 7*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n - 2 + x)*P[n - 2, x] + 3*x*D[P[n - 2, x], x] + (n - 1 + x)*P[n - 1, x] + 3*x*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 + x; v[2] = 3 + 7*x + x^2; for(n = 3, N, my(i = n); v[i] = (n - 2 + x)*v[i - 2] + 3*x*deriv(v[i - 2], x) + (n - 1 + x)*v[i - 1] + 3*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, 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-11c2cb91bf9b2413

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-11c2cb91bf9b2413

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

  • premise (sequence): Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-d88ecdfd26f75232 root: theorem check

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

  • premise (sequence): Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
    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.