A053122 → x ↦ x+1 → D+1
Proved for all n all roots realinterlacing (weak)
Recurrence
\(\left(-10\,x + 2\,x^{2} + 4\,n + 10\,n x - 2\,n x^{2} - 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(4\,n - 4\,n x + 5\,n x^{2} - n x^{3} - n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-6 - 13\,x - x^{2} + 5\,x^{3} - x^{4} + 3\,n + 2\,n x - n x^{2} - 3\,n^{2} - 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 0 | 1 | 1 | |||||||||||||
| 3 | -2 | 0 | 1 | -1 | ||||||||||||
| 4 | 2 | -5 | 0 | 1 | -2 | |||||||||||
| 5 | 1 | 8 | -9 | 0 | 1 | 1 | ||||||||||
| 6 | -4 | 0 | 20 | -14 | 0 | 1 | 3 | |||||||||
| 7 | 3 | -16 | -9 | 40 | -20 | 0 | 1 | -1 | ||||||||
| 8 | 2 | 21 | -36 | -35 | 70 | -27 | 0 | 1 | -4 | |||||||
| 9 | -6 | 0 | 75 | -56 | -90 | 112 | -35 | 0 | 1 | 1 | ||||||
| 10 | 4 | -33 | -36 | 190 | -56 | -189 | 168 | -44 | 0 | 1 | 5 | |||||
| 11 | 3 | 40 | -90 | -176 | 385 | 0 | -350 | 240 | -54 | 0 | 1 | -1 | ||||
| 12 | -8 | 0 | 184 | -140 | -540 | 658 | 168 | -594 | 330 | -65 | 0 | 1 | -6 | |||
| 13 | 5 | -56 | -90 | 560 | -35 | -1296 | 966 | 528 | -945 | 440 | -77 | 0 | 1 | 1 | ||
| 14 | 4 | 65 | -180 | -530 | 1280 | 567 | -2640 | 1200 | 1188 | -1430 | 572 | -90 | 0 | 1 | 7 | |
| 15 | -10 | 0 | 365 | -280 | -1890 | 2288 | 2289 | -4752 | 1155 | 2288 | -2079 | 728 | -104 | 0 | 1 | -1 |
Compute more
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Technical details
- Node id
n-7f8a59ba04fc7524· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-74ec4878e96f8238
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: 0, 1; 3: -2, 0, 1; 4: 2, -5, 0, 1; 5: 1, 8, -9, 0, 1; 6: -4, 0, 20, -14, 0, 1; 7: 3, -16, -9, 40, -20, 0, 1; 8: 2, 21, -36, -35, 70, -27, 0, 1; 9: -6, 0, 75, -56, -90, 112, -35, 0, 1; 10: 4, -33, -36, 190, -56, -189, 168, -44, 0, 1;
%F FormulaThe row polynomials satisfy (2*n^3-6*n^2+4*n+(10*n-10)*x-(2*n-2)*x^2)*P(n,x) = -(n^3-4*n-(n^3-4*n)*x-5*n*x^2+n*x^3)*P(n-1,x) - (3*n^2-3*n+6+(2*n^2-2*n+13)*x-(n^2-n-1)*x^2-5*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*P[n - 1, x] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*v[i - 1] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*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)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
References
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023