A028338 → x ↦ −2x+13 → 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,            1;
  1:          49,          -14,           1;
  2:        -448,          176,         -23,           1;
  3:        4536,        -2223,         407,         -33,         1;
  4:      -50400,        29180,       -6724,         771,       -44,         1;
  5:      609840,      -402974,      110299,      -16010,      1300,       -56,       1;
  6:    -7983360,      5880096,    -1844016,      319234,    -32955,      2029,     -69,       1;
  7:   112432320,    -90739272,    31813774,    -6330107,    781949,    -61418,    2996,     -83,    1;
  8: -1695133440,   1479836064,  -569907352,   127101428, -18093642,   1705305, -106428,    4242,  -98,    1;
  9: 27243216000, -25469564400, 10632128940, -2610196356, 417415949, -45442026, 3411219, -174384, 5811, -114, 1;

The row polynomials satisfy (n+5-x)*P(n,x) = -(n^2+12*n+36-(2*n+12)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -6 + x.

P[0, x_] := -6 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 12*n + 36 - (2*n + 12)*x + x^2)*P[n - 1, x])/(n + 5 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -6 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 12*n + 36 - (2*n + 12)*x + x^2)*v[i - 1])/(n + 5 - 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-e24e4dcabf4aa882

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-e24e4dcabf4aa882

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

  • premise (sequence): Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r =…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-297f51c747adaedd

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-59b849691f316c6f root: theorem check

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

        • parent_provers root_domain e-ef36674c01230a86 root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r =…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-297f51c747adaedd (expanded above)

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.