A035469 → 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:            5,            1;
   2:           32,           13,            1;
   3:          308,          172,           25,           1;
   4:         3920,         2680,          544,          41,          1;
   5:        61880,        49000,        12400,        1320,         61,         1;
   6:      1164800,      1034320,       307840,       41320,       2720,        85,        1;
   7:     25450880,     24814720,      8386840,     1337000,     111860,      5012,      113,       1;
   8:    632952320,    667670080,    250371520,    45702720,    4572960,    261968,     8512,     145,     1;
   9:  17649632000,  19923904000,   8154543040,  1663264960,  191709840,  13203120,   551040,   13584,   181,   1;
  10: 545312768000, 653283030400, 288296982400, 64572334400, 8359718080, 666251040, 33566400, 1066800, 20640, 221, 1;

The row polynomials satisfy P(n,x) = (3*n-5+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3*n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 5 + x, P(2,x) = 32 + 13*x + x^2.

P[1, x_] := 5 + x;
P[2, x_] := 32 + 13*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3*n - 5 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3*n + x)*P[n - 1, x] + 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] = 5 + x; v[2] = 32 + 13*x + x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 5 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3*n + x)*v[i - 1] + 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-d3f71e9704a81a95

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-d3f71e9704a81a95

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

  • premise (sequence): Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-e78bf90f7e218157 root: theorem check

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

  • premise (sequence): Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
    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.