A257626 → 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:        7,           6;
  1:       42,         114,           36;
  2:      252,        1512,         1440,           216;
  3:     1512,       17280,        35100,         16092,           1296;
  4:     9072,      182088,       676188,        658368,         167508,           7776;
  5:    54432,     1825416,     11344212,      19917252,       10867932,        1666980,          46656;
  6:   326592,    17711784,    173965644,     501027120,      492454080,      164923128,       16099236,        279936;
  7:  1959552,   168084072,   2507692932,   11136580164,    17824837320,    10818091728,     2360481588,     152311428,      1679616;
  8: 11757312,  1570703400,  34582427388,  226743541056,   553269998124,   546581740968,   218433142260,   32390029872,   1420351524,    10077696;
  9: 70543872, 14519283048, 461654136180, 4327845368292, 15368740468164, 23044857266052, 15048902268252, 4143994653420, 430834809708, 13110688836, 60466176;

The row polynomials satisfy P(n,x) = (6+3*n*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (5+(3*n+6)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 7 + 6*x, P(1,x) = 42 + 114*x + 36*x^2.

P[0, x_] := 7 + 6*x;
P[1, x_] := 42 + 114*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 3*n*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (5 + (3*n + 6)*x)*P[n - 1, x] + (3*x - 3*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] = 7 + 6*x; v[2] = 42 + 114*x + 36*x^2; for(n = 2, N, my(i = n + 1); v[i] = (6 + 3*n*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (5 + (3*n + 6)*x)*v[i - 1] + (3*x - 3*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-0e437c53d6b0d63a

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-0e437c53d6b0d63a

  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) = 3*x + 6.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-5ba7ec4213cfb517 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) = 3*x + 6.
    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.