A075500 → 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:       150,          190,            31,             1;
  1:       750,         2050,           655,            51,             1;
  2:      3750,        21250,         11875,          1675,            76,            1;
  3:     18750,       216250,        199375,         45375,          3575,          106,           1;
  4:     93750,      2181250,       3206875,       1106875,        134750,         6755,         141,          1;
  5:    468750,     21906250,      50284375,      25344375,       4475625,       337400,       11690,        181,         1;
  6:   2343750,    219531250,     776171875,     557171875,     137235000,     14597625,      746550,      18930,       226,       1;
  7:  11718750,   2197656250,   11862109375,   11919609375,    3988046875,    575163750,    40726875,    1503750,     29100,     276,     1;
  8:  58593750,  21988281250,  180129296875,  250254296875,  111620781250,  21242959375,  2000604375,  100876875,   2813250,   42900,   331,   1;
  9: 292968750, 219941406250, 2723927734375, 5185215234375, 3040773828125, 748909562500, 91264112500, 6035679375, 227473125, 4958250, 61105, 391, 1;

The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 150 + 190*x + 31*x^2 + x^3.

P[0, x_] := 150 + 190*x + 31*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + 5*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] = 150 + 190*x + 31*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + 5*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-206745053438cac1

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-206745053438cac1

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 5).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-42338bfb8394e6f8 root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 5).
    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.