A225471 → 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:            24,            11,            1;
  2:           252,           141,           22,           1;
  3:          3696,          2327,          467,          37,           1;
  4:         69300,         47385,        11116,        1166,          56,         1;
  5:       1580040,       1150371,       301269,       37790,        2450,        79,        1;
  6:      42397740,      32459301,      9241954,     1317079,      103720,      4579,      106,       1;
  7:    1308273120,    1044215343,    317797479,    49924763,     4522819,    245357,     7861,     137,     1;
  8:   45626025060,   37730393649,  12131321688,  2060042508,   207818128,  13092310,   520072,   12652,   172,   1;
  9: 1774345419000, 1513059932475, 509616554985, 92278402124, 10147263524, 717454794, 33344094, 1012956, 19356, 211, 1;

The row polynomials satisfy (4*n+x)*P(n,x) = (16*n^2+12*n-4+(8*n+3)*x+x^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^2 + 12*n - 4 + (8*n + 3)*x + x^2)*P[n - 1, x])/(4*n + x)]];
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^2 + 12*n - 4 + (8*n + 3)*x + x^2)*v[i - 1])/(4*n + 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-6a5312030eb510ce

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-6a5312030eb510ce

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

  • premise (sequence): Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-dad3570ee638379f root: theorem check

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

    • parent_provers root_domain e-381fad74fe44e78a root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
    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.