A114655 → 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:
   1:    6,     2;
   2:   12,    14,      2;
   3:   24,    60,     26,       2;
   4:   48,   208,    184,      42,       2;
   5:   96,   640,    960,     440,      62,       2;
   6:  192,  1824,   4160,    3200,     900,      86,      2;
   7:  384,  4928,  15904,   18480,    8680,    1652,    114,     2;
   8:  768, 12800,  55552,   90944,   64288,   20384,   2800,   146,    2;
   9: 1536, 32256, 181248,  397824,  395136,  188160,  43008,  4464,  182,   2;
  10: 3072, 79360, 560640, 1589760, 2112768, 1411200, 483840, 83520, 6780, 222, 2;

The row polynomials satisfy (9*n^2+15*n-6-(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (18*n^2+30*n-12+(23*n^2-21*n+10)*x-(4*n^2-12*n+8)*x^2+(3*n^2-3*n)*x^3)*P(n-1,x) + (36*n*x-(26*n-20)*x^2+(8*n-14)*x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 6 + 2*x.

P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*P[n - 1, x] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; for(n = 2, N, my(i = n); v[i] = ((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*v[i - 1] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-798bfa3a83d1e4b4

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-798bfa3a83d1e4b4

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-784ea0037cdd0731 root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    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.