A257624 → 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:        6,          5;
  1:       30,         85,           25;
  2:      150,        995,          940,           125;
  3:      750,       9985,        21045,          9195,           625;
  4:     3750,      92255,       368540,        357480,         83810,          3125;
  5:    18750,     811165,      5595065,      10103890,       5329840,        731105,         15625;
  6:    93750,    6911195,     77519460,     235525335,     231016510,      73000785,       6198840,         78125;
  7:   468750,   57680185,   1009487605,    4832003925,    7901944925,    4678506875,     943398015,      51575095,       390625;
  8:  2343750,  474800855,  12585498820,   90649451450,  230295832660,  227225891660,   86980432180,   11699780990,    423694510,    1953125;
  9: 11718750, 3872234965, 152055220905, 1593503645220, 5986147836360, 9127405961190, 5849571396150, 1519108711860, 140830240770, 3450884205, 9765625;

The row polynomials satisfy P(n,x) = (5+(3*n-1)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (4+(3*n+5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 5*x, P(1,x) = 30 + 85*x + 25*x^2.

P[0, x_] := 6 + 5*x;
P[1, x_] := 30 + 85*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + (3*n - 1)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (4 + (3*n + 5)*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] = 6 + 5*x; v[2] = 30 + 85*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (5 + (3*n - 1)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (4 + (3*n + 5)*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-148e9e46774006f0

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-148e9e46774006f0

  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 + 5.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-013c0ad711a68b34 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 + 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.