A028338 → x ↦ (−1/2)x+1 (shift +2) → 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:             -352,               164,              -23,                1;
  1:             5760,             -3024,              536,              -39,              1;
  2:          -116736,             66944,           -13840,             1320,            -59,              1;
  3:          2826240,          -1736192,           401344,           -45680,           2740,            -83,            1;
  4:        -79626240,          51720192,        -13031424,          1685824,        -122640,           5068,         -111,           1;
  5:       2559836160,       -1741897728,        470388736,        -67165952,        5621504,        -285152,         8624,        -143,         1;
  6:     -92484403200,       65482260480,     -18729680896,       2895294464,     -270041856,       15907584,      -596064,       13776,      -179,        1;
  7:    3709945774080,    -2718956519424,     816604839936,    -134817054720,    13719956480,     -907506432,     39785088,    -1147680,     20940,     -219,     1;
  8: -163618121318400,   123613143367680,  -38726423543808,    6760412020736,  -739598295040,    53715031040,  -2660469504,    90338688,  -2069760,    30580,  -263,    1;
  9: 7868890113638400, -6108193939783680, 1985824746897408, -363777325531136, 42317113163776, -3321614602240, 181579126784, -7001372928, 189771648, -3538480, 43208, -311, 1;

The row polynomials satisfy (4*n+7-x)*P(n,x) = -(16*n^2+76*n+88-(8*n+19)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -352 + 164*x - 23*x^2 + x^3.

P[0, x_] := -352 + 164*x - 23*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 + 76*n + 88 - (8*n + 19)*x + x^2)*P[n - 1, x])/(4*n + 7 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -352 + 164*x - 23*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^2 + 76*n + 88 - (8*n + 19)*x + x^2)*v[i - 1])/(4*n + 7 - 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-f961be0564a72cf6

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-f961be0564a72cf6

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

  • premise (sequence): Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >=…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 2 e-41f01d4be10aeae3

      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 >= 2, 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): Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >=…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 2 e-41f01d4be10aeae3 (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.