A028338 → x ↦ (−1/5)x+1 → 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:                -9,                  1;
  1:               190,                -29,                1;
  2:             -5800,               1070,              -59,               1;
  3:            234000,             -48900,             3440,             -99,             1;
  4:         -11760000,            2690000,          -221500,            8400,          -149,            1;
  5:         708000000,         -173660000,         16015000,         -726500,         17350,         -209,          1;
  6:      -49680000000,        12891600000,      -1296960000,        66955000,      -1942500,        31990,       -279,         1;
  7:     3981600000000,     -1082772000000,     116810800000,     -6660710000,     222530000,     -4503800,      54320,      -359,      1;
  8:  -358848000000000,    101561760000000,  -11608876000000,    716951600000,  -26708010000,    628194000,   -9395400,     86640,   -449,    1;
  9: 35925120000000000, -10525982400000000, 1263630600000000, -83371320000000, 3389997500000, -89572770000, 1568280000, -18063000, 131550, -549, 1;

The row polynomials satisfy (10*n-1-x)*P(n,x) = -(100*n^2+90*n-(20*n+9)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -9 + x.

P[0, x_] := -9 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(100*n^2 + 90*n - (20*n + 9)*x + x^2)*P[n - 1, x])/(10*n - 1 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -9 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(100*n^2 + 90*n - (20*n + 9)*x + x^2)*v[i - 1])/(10*n - 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-cdcdb3aca64a476d

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-cdcdb3aca64a476d

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

  • premise (sequence): Triangle read by rows: T(n, m) = S1(n, m)*10^(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 0 e-040c826eba4e7d74

      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): Triangle read by rows: T(n, m) = S1(n, m)*10^(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 0 e-040c826eba4e7d74 (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.