A035342 → 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:           4,           1;
  1:          18,          10,           1;
  2:         120,          96,          19,          1;
  3:        1050,        1062,         303,         31,          1;
  4:       11340,       13620,        4965,        735,         46,         1;
  5:      145530,      200070,       87525,      16665,       1515,        64,       1;
  6:     2162160,     3321360,     1678635,     386085,      45360,      2793,      85,      1;
  7:    36486450,    61571790,    35049735,    9368415,    1335705,    106680,    4746,    109,     1;
  8:   689188500,  1261654380,   794424645,  240414615,   40001850,   3890565,  225162,   7578,   136,   1;
  9: 14404039650, 28328709150, 19466513325, 6546378825, 1237877235, 140948640, 9960615, 437130, 11520, 166, 1;

The row polynomials satisfy (8*n^3+24*n^2+22*n+6-x^2)*P(n,x) = (16*n^4+72*n^3+108*n^2+62*n+12+(8*n^3+24*n^2+26*n+8)*x-(2*n+3)*x^2-x^3)*P(n-1,x) + ((8*n^3+24*n^2+26*n+8)*x-4*x^2-x^3)*d/dx P(n-1,x) - 2*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4 + x.

P[0, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 + 72*n^3 + 108*n^2 + 62*n + 12 + (8*n^3 + 24*n^2 + 26*n + 8)*x - (2*n + 3)*x^2 - x^3)*P[n - 1, x] + ((8*n^3 + 24*n^2 + 26*n + 8)*x - 4*x^2 - x^3)*D[P[n - 1, x], x] - 2*x^2*D[P[n - 1, x], {x, 2}])/(8*n^3 + 24*n^2 + 22*n + 6 - x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^4 + 72*n^3 + 108*n^2 + 62*n + 12 + (8*n^3 + 24*n^2 + 26*n + 8)*x - (2*n + 3)*x^2 - x^3)*v[i - 1] + ((8*n^3 + 24*n^2 + 26*n + 8)*x - 4*x^2 - x^3)*deriv(v[i - 1], x) - 2*x^2*deriv(deriv(v[i - 1], x), x))/(8*n^3 + 24*n^2 + 22*n + 6 - x^2)); 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-91082a28c5d8ee02

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-91082a28c5d8ee02

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

  • premise (sequence): The convolution matrix of the double factorial of odd numbers (A001147).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-cf2bc165f5c18b21 root: theorem check

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

  • premise (sequence): The convolution matrix of the double factorial of odd numbers (A001147).
    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.