A344565 → 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:   18,    15;
  1:   60,   105,     45;
  2:  150,   405,    360,    105;
  3:  315,  1155,   1575,    945,    210;
  4:  588,  2730,   5040,   4620,   2100,     378;
  5: 1008,  5670,  13230,  16380,  11340,    4158,    630;
  6: 1620, 10710,  30240,  47250,  44100,   24570,   7560,    990;
  7: 2475, 18810,  62370, 117810, 138600,  103950,  48510,  12870,   1485;
  8: 3630, 31185, 118800, 263340, 374220,  353430, 221760,  89100,  20790,  2145;
  9: 5148, 49335, 212355, 540540, 900900, 1027026, 810810, 437580, 154440, 32175, 3003;

The row polynomials satisfy (2*n^2+6*n+4+(n^2+5*n+4)*x)*P(n,x) = (2*n^2+14*n+24+(3*n^2+23*n+44)*x+(n^2+9*n+20)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 18 + 15*x.

P[0, x_] := 18 + 15*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 14*n + 24 + (3*n^2 + 23*n + 44)*x + (n^2 + 9*n + 20)*x^2)*P[n - 1, x])/(2*n^2 + 6*n + 4 + (n^2 + 5*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 18 + 15*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 14*n + 24 + (3*n^2 + 23*n + 44)*x + (n^2 + 9*n + 20)*x^2)*v[i - 1])/(2*n^2 + 6*n + 4 + (n^2 + 5*n + 4)*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-aeab44183cd7972d

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-aeab44183cd7972d

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

  • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-e2cde2e22920ce64 root: theorem check

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

    • preserver_image reversal, shift 0 e-4dae17a76888d380

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-e2cde2e22920ce64 (expanded above)
  • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A178821 → Pₙ + Pₙ₊₁

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.