A178822 → 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:    7,     6;
  1:   27,    48,     21;
  2:   77,   210,    189,     56;
  3:  182,   672,    924,    560,    126;
  4:  378,  1764,   3276,   3024,   1386,     252;
  5:  714,  4032,   9450,  11760,   8190,    3024,    462;
  6: 1254,  8316,  23562,  36960,  34650,   19404,   6006,    792;
  7: 2079, 15840,  52668,  99792, 117810,   88704,  41580,  11088,   1287;
  8: 3289, 28314, 108108, 240240, 342342,  324324, 204204,  82368,  19305,  2002;
  9: 5005, 48048, 207207, 528528, 882882, 1009008, 798798, 432432, 153153, 32032, 3003;

The row polynomials satisfy (2*n^2+7*n+5+(n^2+6*n+5)*x)*P(n,x) = (2*n^2+17*n+35+(3*n^2+28*n+65)*x+(n^2+11*n+30)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 6*x.

P[0, x_] := 7 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 17*n + 35 + (3*n^2 + 28*n + 65)*x + (n^2 + 11*n + 30)*x^2)*P[n - 1, x])/(2*n^2 + 7*n + 5 + (n^2 + 6*n + 5)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 7 + 6*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 17*n + 35 + (3*n^2 + 28*n + 65)*x + (n^2 + 11*n + 30)*x^2)*v[i - 1])/(2*n^2 + 7*n + 5 + (n^2 + 6*n + 5)*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-c021bdf930303dcb

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-c021bdf930303dcb

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

  • premise (sequence): Triangle read by rows: T(n,k) = C(n+5,5) * 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-548a05679c99d179 root: theorem check

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

    • preserver_image reversal, shift 0 e-0d5dad6bd99a01a4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = C(n+5,5) * 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-548a05679c99d179 (expanded above)
  • premise (sequence): Triangle read by rows: T(n,k) = C(n+5,5) * 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 ?

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.