A178821 → Pₙ + Pₙ₊₁

OEIS matches ?

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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:    6,     5;
  1:   20,    35,    15;
  2:   50,   135,   120,     35;
  3:  105,   385,   525,    315,     70;
  4:  196,   910,  1680,   1540,    700,    126;
  5:  336,  1890,  4410,   5460,   3780,   1386,    210;
  6:  540,  3570, 10080,  15750,  14700,   8190,   2520,    330;
  7:  825,  6270, 20790,  39270,  46200,  34650,  16170,   4290,   495;
  8: 1210, 10395, 39600,  87780, 124740, 117810,  73920,  29700,  6930,   715;
  9: 1716, 16445, 70785, 180180, 300300, 342342, 270270, 145860, 51480, 10725, 1001;

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) = 6 + 5*x.

P[0, x_] := 6 + 5*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] = 6 + 5*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-19ec7bcbd5a09ffd

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-19ec7bcbd5a09ffd

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(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-20c79d8930f25d5b root: theorem check

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

    • preserver_image reversal, shift 0 e-12f9ed6b8198f9ce

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(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-20c79d8930f25d5b (expanded above)
  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(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.

• A344565 → 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.