A257616 → 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:    3,         2;
  1:    6,        34,            4;
  2:   12,       344,          316,             8;
  3:   24,      2968,         9048,          2664,             16;
  4:   48,     24320,       184160,        178080,          21680,             32;
  5:   96,    196000,      3194624,       7192384,        3040864,         174368,             64;
  6:  192,   1571456,     50866752,     225855360,      221857600,       48021888,        1397184,           128;
  7:  384,  12579712,    770803840,    6126728064,    11598435200,     5955737216,      724353408,      11182720,          256;
  8:  768, 100656128,  11336378368,  151538021888,   496301582848,   490829875712,   146353088512,   10624604672,     89473792,       512;
  9: 1536, 805290496, 163675008000, 3525223348224, 18626838647808, 31519929093120, 18287546793984, 3390206638080, 153151059456, 715817472, 1024;

The row polynomials satisfy P(n,x) = (2+(6*n-10)*x)*P(n-2,x) + (6*x-6*x^2)*d/dx P(n-2,x) + (1+(6*n+2)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 34*x + 4*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 6 + 34*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (6*n - 10)*x)*P[n - 2, x] + (6*x - 6*x^2)*D[P[n - 2, x], x] + (1 + (6*n + 2)*x)*P[n - 1, x] + (6*x - 6*x^2)*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] = 3 + 2*x; v[2] = 6 + 34*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (6*n - 10)*x)*v[i - 2] + (6*x - 6*x^2)*deriv(v[i - 2], x) + (1 + (6*n + 2)*x)*v[i - 1] + (6*x - 6*x^2)*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-4b3d723d1668996c

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-4b3d723d1668996c

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-d30c56e449c0455e root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.
    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.