A075504 → 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:          810,             594,              55,                1;
  1:         7290,           11502,            2079,               91,               1;
  2:        65610,          214326,           67635,             5355,             136,              1;
  3:       590490,         3923478,         2040471,           260415,           11475,            190,             1;
  4:      5314410,        71213094,        59016195,         11415411,          776790,          21735,           253,            1;
  5:     47829690,      1287150102,      1664650359,        469970991,        46370961,        1950480,         37674,          325,          1;
  6:    430467210,     23216531526,     46232709795,      18583606035,      2556664236,      151696881,       4323942,        61074,        406,        1;
  7:   3874204890,    418328034678,   1271499695991,     715242527055,    133633496655,    10748295810,     424105227,      8721270,      93960,      496,      1;
  8:  34867844010,   7533778829094,  34748819826435,   27020230669971,   6728749876530,   714041470395,   37466925111,   1052036667,   16332030,   138600,    595,   1;
  9: 313810596090, 135642886767702, 945751914142839, 1007477123945391, 329813975113821, 45286989277860, 3074457752388, 113213565135, 2374931097, 28806030, 197505, 703, 1;

The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 810 + 594*x + 55*x^2 + x^3.

P[0, x_] := 810 + 594*x + 55*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(9 + x)*P[n - 1, x] + 9*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] = 810 + 594*x + 55*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (9 + 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 - 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-cdccc64d5381f9ff

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-cdccc64d5381f9ff

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 9).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-b7566ed62ccee4c1 root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 9).
    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.