A156992 → 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:        3,         2;
  1:        8,        14,          6;
  2:       30,        84,         78,         24;
  3:      144,       552,        792,        504,        120;
  4:      840,      4080,       7920,       7680,       3720,         720;
  5:     5760,     33840,      82800,     108000,      79200,       30960,       5040;
  6:    45360,    312480,     922320,    1512000,    1486800,      876960,     287280,      40320;
  7:   403200,   3185280,   11007360,   21732480,   26812800,    21168000,   10442880,    2943360,     362880;
  8:  3991680,  35562240,  140797440,  325140480,  482630400,   477550080,  314979840,  133539840,   33022080,   3628800;
  9: 43545600, 431827200, 1926892800, 5094835200, 8839756800, 10516262400, 8687347200, 4920652800, 1828915200, 402796800, 39916800;

The row polynomials satisfy (n+2+(n+1)*x)*P(n,x) = (n^2+4*n+3+(2*n^2+7*n+5)*x+(n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.

P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(n + 2 + (n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*v[i - 1])/(n + 2 + (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-107bbdcb60a21a66

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-107bbdcb60a21a66

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

  • premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-d07da0cb5a6ac36b root: theorem check

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

    • parent_provers root_domain e-e811f0fabc9e8e88 root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    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.