A166979 → 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,          122,             31,              1;
   2: 2,         1222,            711,             59,              1;
   3: 2,        12222,          14731,           2363,             96,             1;
   4: 2,       122222,         292111,          80895,           5915,           142,            1;
   5: 2,      1222222,        5672331,        2557171,         299750,         12447,          197,           1;
   6: 2,     12222222,      108996511,       77273119,       13647921,        872312,        23282,         261,          1;
   7: 2,    122222222,     2083155931,     2272643843,      582246196,      53774273,      2152822,       39986,        334,        1;
   8: 2,   1222222222,    39702184911,    65717183535,    23815753095,    3055862754,    172179483,     4711926,      64368,      416,      1;
   9: 2,  12222222222,   755563735531,  1879783323891,   946900048050,  164385439779,  12525734319,   473742747,    9410790,    98480,    507,   1;
  10: 2, 122222222222, 14367933197311, 53389496804479, 36915085101741, 8508630277884, 853300827324, 42845270127, 1160730417, 17486150, 144617, 607, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + 122*x + 31*x^2 + x^3.

P[1, x_] := 2 + 122*x + 31*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 9*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 + 122*x + 31*x^2 + x^3; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 9*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-a20ccc97cc2b4079

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-a20ccc97cc2b4079

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

  • premise (sequence): Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-b6e553c117a424c0 root: theorem check

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

  • premise (sequence): Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m…
    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.