A075499 → 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:       80,         124,           25,            1;
  1:      320,        1072,          424,           41,            1;
  2:     1280,        8896,         6160,         1080,           61,            1;
  3:     5120,       72448,        82816,        23440,         2300,           85,           1;
  4:    20480,      584704,      1066240,       457856,        69440,         4340,         113,          1;
  5:    81920,     4698112,     13379584,      8391936,      1846656,       173600,        7504,        145,        1;
  6:   327680,    37666816,    165253120,    147650560,     45325056,      6013056,      383712,      12144,      181,       1;
  7:  1310720,   301662208,   2020704256,   2527662080,   1054151680,    189638400,    16756992,     772320,    18660,     221,     1;
  8:  5242880,  2414608384,  24550113280,  42463297536,  23610695680,   5605473280,   658834176,   41471232,  1444080,   27500,   265,   1;
  9: 20971520, 19322109952, 297015967744, 703962873856, 514677211136, 158142054400, 24052830208, 1985913600, 93458112, 2544080, 39160, 313, 1;

The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 80 + 124*x + 25*x^2 + x^3.

P[0, x_] := 80 + 124*x + 25*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + 4*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] = 80 + 124*x + 25*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (4 + x)*v[i - 1] + 4*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-4d98a7f5f8c29383

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-4d98a7f5f8c29383

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 4).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-6592d91fe2aba204 root: theorem check

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

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 4).
    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.