Real-rooted atlas

A334218 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(\left(-1 - 2\,x + n + 5\,n x - 4\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(6\,x - 5\,n x - 6\,n x^{2} + n^{2} x + 11\,n^{2} x^{2} - 6\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{2} + n x + n x^{2} - 2\,n x^{3} - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{2} - n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(3 - 2\,x - n - 4\,n x - 2\,n x^{2} + 5\,n^{2} x + 5\,n^{2} x^{2} - n^{3} x - 4\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n-1}(x) + \left(n x + n x^{2} - 2\,n x^{3} - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{2} - n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 5\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
2123
30538
40719430
50971595144
60112114511616840
70135552510227040575760
8015135911642214429927967845360
90173183481651598991500993959722319403200
100197243184717102291117024719209771483035029103991680
1102116163672320588945215975794152962345153192531260111531143545600
12023355752357884314489681312712622031048701218644802696809618427872444312518918400
1302577511804916715888176197899753023013506942229517086888582018134271549624195553091136706022400
140271675712693368776990820567880816522319136083033965067774219408412746058381395643685823207692431145051493405312000
150293600278877587436136300904450207411521372764405745142004943644064634249219857488443539213600925299687795468174202245549151394852659200
160317695672892555341654561638227923580496118370809257935401769215956758018872093251748505175211683008115473243635834126136375143022214413595240471622230464256000

Computed in your browser ?

Technical details
Node id
n-ffeb6f77baa8bcee · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-a2866d6de87d0de1

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): 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.

The atlas's own rows: n = 2, …, ascending powers of x
%e Example
Triangle begins:
   2: 1,  2;
   3: 0,  5,     3;
   4: 0,  7,    19,      4;
   5: 0,  9,    71,     59,       5;
   6: 0, 11,   211,    451,     161,        6;
   7: 0, 13,   555,   2510,    2270,      405,        7;
   8: 0, 15,  1359,  11642,   21442,     9927,      967,       8;
   9: 0, 17,  3183,  48165,  159899,   150099,    39597,    2231,      9;
  10: 0, 19,  7243, 184717, 1022911,  1702471,   920977,  148303,   5029,    10;
  11: 0, 21, 16163, 672320, 5889452, 15975794, 15296234, 5153192, 531260, 11153, 11;
%F Formula
The row polynomials satisfy (n-1+(n^3-4*n^2+5*n-2)*x)*P(n,x) = ((n^2-5*n+6)*x+(n^4-6*n^3+11*n^2-6*n)*x^2)*P(n-2,x) + ((n-2)*x+(n^3-3*n^2+n+2)*x^2-(n^3-3*n^2+2*n)*x^3)*d/dx P(n-2,x) - (n-3+(n^3-5*n^2+4*n+2)*x-(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-1,x) + (n*x+(n^3-3*n^2+n)*x^2-(n^3-3*n^2+2*n)*x^3)*d/dx P(n-1,x) for n >= 4, with P(2,x) = 1 + 2*x, P(3,x) = 5*x + 3*x^2.
%t Mathematica
P[2, x_] := 1 + 2*x;
P[3, x_] := 5*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 - 5*n + 6)*x + (n^4 - 6*n^3 + 11*n^2 - 6*n)*x^2)*P[n - 2, x] + ((n - 2)*x + (n^3 - 3*n^2 + n + 2)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*D[P[n - 2, x], x] - (n - 3 + (n^3 - 5*n^2 + 4*n + 2)*x - (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 1, x] + (n*x + (n^3 - 3*n^2 + n)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(n - 1 + (n^3 - 4*n^2 + 5*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1 + 2*x; v[2] = 5*x + 3*x^2; for(n = 4, N, my(i = n - 1); v[i] = (((n^2 - 5*n + 6)*x + (n^4 - 6*n^3 + 11*n^2 - 6*n)*x^2)*v[i - 2] + ((n - 2)*x + (n^3 - 3*n^2 + n + 2)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*deriv(v[i - 2], x) - (n - 3 + (n^3 - 5*n^2 + 4*n + 2)*x - (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 1] + (n*x + (n^3 - 3*n^2 + n)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(n - 1 + (n^3 - 4*n^2 + 5*n - 2)*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(11)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 2, 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-33c9a461744a57dc

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

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.