A178820 → 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:   5,    4;
  1:  14,   24,    10;
  2:  30,   80,    70,    20;
  3:  55,  200,   270,   160,    35;
  4:  91,  420,   770,   700,   315,    56;
  5: 140,  784,  1820,  2240,  1540,   560,    84;
  6: 204, 1344,  3780,  5880,  5460,  3024,   924,   120;
  7: 285, 2160,  7140, 13440, 15750, 11760,  5460,  1440,   165;
  8: 385, 3300, 12540, 27720, 39270, 36960, 23100,  9240,  2145,  220;
  9: 506, 4840, 20790, 52800, 87780, 99792, 78540, 42240, 14850, 3080, 286;

The row polynomials satisfy (2*n^2+5*n+3+(n^2+4*n+3)*x)*P(n,x) = (2*n^2+11*n+15+(3*n^2+18*n+27)*x+(n^2+7*n+12)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 5 + 4*x.

P[0, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 11*n + 15 + (3*n^2 + 18*n + 27)*x + (n^2 + 7*n + 12)*x^2)*P[n - 1, x])/(2*n^2 + 5*n + 3 + (n^2 + 4*n + 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 5 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 11*n + 15 + (3*n^2 + 18*n + 27)*x + (n^2 + 7*n + 12)*x^2)*v[i - 1])/(2*n^2 + 5*n + 3 + (n^2 + 4*n + 3)*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-ac667fe3dc6003e4

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-ac667fe3dc6003e4

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

  • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-387009a21707feb6 root: theorem check

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

    • preserver_image reversal, shift 0 e-17b3b03bdb024482

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-387009a21707feb6 (expanded above)
  • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
    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.

• A007318 → θ+1+n → D

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.