A049374 → 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:           7,            1;
  1:          48,           19,            1;
  2:         378,          294,           37,            1;
  3:        3360,         4476,          996,           61,           1;
  4:       33264,        70224,        24360,         2520,          91,          1;
  5:      362880,      1152648,       580824,        89820,        5340,        127,        1;
  6:     4324320,     19891872,     13907376,      3003504,      260610,      10038,      169,       1;
  7:    55883520,    361291392,    339204096,     97918128,    11598384,     641928,    17304,     217,     1;
  8:   778377600,   6901614720,   8488839744,   3175913664,   492007824,   36623664,  1403136,   27936,   271,   1;
  9: 11623772160, 138468185856, 218796754176, 103671247680, 20384844480, 1956322368, 99747648, 2799720, 42840, 331, 1;

The row polynomials satisfy P(n,x) = (n+4+x)*P(n-2,x) + 5*x*d/dx P(n-2,x) + (n+5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 7 + x, P(1,x) = 48 + 19*x + x^2.

P[0, x_] := 7 + x;
P[1, x_] := 48 + 19*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + x)*P[n - 2, x] + 5*x*D[P[n - 2, x], x] + (n + 5 + x)*P[n - 1, x] + 5*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] = 7 + x; v[2] = 48 + 19*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + 4 + x)*v[i - 2] + 5*x*deriv(v[i - 2], x) + (n + 5 + x)*v[i - 1] + 5*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-0a2b33f6b07eac20

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-0a2b33f6b07eac20

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

  • premise (sequence): A triangle of numbers related to triangle A030527.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-eb06c12f54622221 root: theorem check

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

  • premise (sequence): A triangle of numbers related to triangle A030527.
    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.