A110162 → x ↦ x+1
Proved for all n all roots real, in (−1, 3)interlacing (weak)
Recurrence
\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(2 - 2\,x - n + n x\right)\,P_{n-1}(x) + \left(-3 - 2\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -1 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | -1 | 1 | 0 | |||||||||||||
| 3 | -1 | -2 | 1 | -2 | ||||||||||||
| 4 | 2 | 0 | -3 | 1 | 0 | |||||||||||
| 5 | -1 | 4 | 2 | -4 | 1 | 2 | ||||||||||
| 6 | -1 | -5 | 5 | 5 | -5 | 1 | 0 | |||||||||
| 7 | 2 | 0 | -12 | 4 | 9 | -6 | 1 | -2 | ||||||||
| 8 | -1 | 7 | 7 | -21 | 0 | 14 | -7 | 1 | 0 | |||||||
| 9 | -1 | -8 | 12 | 24 | -30 | -8 | 20 | -8 | 1 | 2 | ||||||
| 10 | 2 | 0 | -27 | 9 | 54 | -36 | -21 | 27 | -9 | 1 | 0 | |||||
| 11 | -1 | 10 | 15 | -60 | -15 | 98 | -35 | -40 | 35 | -10 | 1 | -2 | ||||
| 12 | -1 | -11 | 22 | 66 | -99 | -77 | 154 | -22 | -66 | 44 | -11 | 1 | 0 | |||
| 13 | 2 | 0 | -48 | 16 | 180 | -120 | -196 | 216 | 9 | -100 | 54 | -12 | 1 | 2 | ||
| 14 | -1 | 13 | 26 | -130 | -65 | 377 | -78 | -390 | 273 | 65 | -143 | 65 | -13 | 1 | 0 | |
| 15 | -1 | -14 | 35 | 140 | -245 | -322 | 651 | 96 | -672 | 308 | 154 | -196 | 77 | -14 | 1 | -2 |
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Technical details
- Node id
n-c9d79575c6360af4· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-a6760e5b116a31f3
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -1, 1; 3: -1, -2, 1; 4: 2, 0, -3, 1; 5: -1, 4, 2, -4, 1; 6: -1, -5, 5, 5, -5, 1; 7: 2, 0, -12, 4, 9, -6, 1; 8: -1, 7, 7, -21, 0, 14, -7, 1; 9: -1, -8, 12, 24, -30, -8, 20, -8, 1; 10: 2, 0, -27, 9, 54, -36, -21, 27, -9, 1;
%F FormulaThe row polynomials satisfy (2*n-4)*P(n,x) = -(n-2-(n-2)*x)*P(n-1,x) - (3+2*x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 - (n - 2)*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 - (n - 2)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (-1, 3)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
Proof chains ? JSON
References
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023