A075502 → 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:         392,           364,             43,               1;
  1:        2744,          5488,           1267,              71,              1;
  2:       19208,         79576,          32095,            3255,            106,             1;
  3:      134456,       1133272,         753571,          123235,           6965,           148,            1;
  4:      941192,      16000264,       16958263,         4204151,         367010,         13181,          197,           1;
  5:     6588344,     224944888,      372123787,       134674491,       17049501,        920612,        22834,         253,         1;
  6:    46118408,    3155816776,     8039544415,      4143009535,      731407026,      55715205,      2039478,       37002,       316,        1;
  7:   322828856,   44227553272,   171986249491,    124043811395,    29742255445,    3071445636,    155649627,     4111590,     56910,      386,      1;
  8:  2259801992,  619508574664,  3655938792583,   3645212968551,  1165022751970,  158742972157,  10698277359,   385898667,   7696920,    83930,    463,   1;
  9: 15818613944, 8675379847288, 77394223218907, 105721901912011, 44421009287501, 7832227582564, 682958562748, 32308602711, 870804627, 13572020, 119581, 547, 1;

The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 392 + 364*x + 43*x^2 + x^3.

P[0, x_] := 392 + 364*x + 43*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + 7*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] = 392 + 364*x + 43*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + 7*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-a22896546ec845f5

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-a22896546ec845f5

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

  • premise (sequence): Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-4e6146a9e968e4df root: theorem check

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

  • premise (sequence): Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).
    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.