A375853 → 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:   10,     8;
   2:   28,    64,     20;
   3:   60,   272,    236,      40;
   4:  110,   832,   1404,     656,      70;
   5:  182,  2072,   5764,    5192,    1526,     112;
   6:  280,  4480,  18616,   27456,   15496,    3136,     168;
   7:  408,  8736,  50760,  111280,  102440,   39744,    5880,     240;
   8:  570, 15744, 121992,  372640,  508300,  320960,   90984,   10272,    330;
   9:  770, 26664, 265848, 1080112, 2047820, 1912160,  881144,  190608,  16962,   440;
  10: 1012, 42944, 535788, 2795584, 7036232, 9116352, 6204184, 2180288, 371844, 26752, 572;

The row polynomials satisfy (4*n^2+2*n-5*n*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+14*n+12+(12*n^2+n-14)*x+(n^2-6*n)*x^2+(3*n^2+7*n+2)*x^3)*P(n-1,x) + ((8*n+8)*x-(8*n+12)*x^2+(2*n+8)*x^3-(2*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 10 + 8*x.

P[1, x_] := 10 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*P[n - 1, x] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 10 + 8*x; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*v[i - 1] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*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-01a8bab2955d58ac

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-01a8bab2955d58ac

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

  • premise (sequence): Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-b3e0ac60e2e16f0a root: theorem check

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

  • premise (sequence): Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
    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.