A179900 → D+1
Proved for all n all roots realinterlacing (weak)
Recurrence
\(\left(-74\,n + 26\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-159 + 91\,x - 17\,x^{2} + x^{3} - 149\,n + 89\,n x - 17\,n x^{2} + n x^{3} + 15\,n^{2} - 3\,n^{2} x + 5\,n^{3} - n^{3} x\right)\,P_{n-1}(x) + \left(546 - 491\,x + 157\,x^{2} - 21\,x^{3} + x^{4} - 21\,n + 10\,n x - n x^{2} - 21\,n^{2} + 10\,n^{2} x - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | -1 | 3 | ||||||
| 2 | 14 | -8 | 1 | 7 | |||||
| 3 | 42 | -43 | 12 | -1 | 10 | ||||
| 4 | 81 | -176 | 87 | -16 | 1 | -23 | |||
| 5 | -188 | -448 | 452 | -146 | 20 | -1 | -311 | ||
| 6 | -3661 | 952 | 1431 | -920 | 220 | -24 | 1 | -2001 | |
| 7 | -30766 | 25179 | -2880 | -3485 | 1630 | -309 | 28 | -1 | -10604 |
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Technical details
- Node id
n-a6a369db31b832e1· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-fc7fdb4b275e3246
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 0)
- 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 4, -1; 2: 14, -8, 1; 3: 42, -43, 12, -1; 4: 81, -176, 87, -16, 1; 5: -188, -448, 452, -146, 20, -1; 6: -3661, 952, 1431, -920, 220, -24, 1; 7: -30766, 25179, -2880, -3485, 1630, -309, 28, -1; 8: -210774, 247080, -99285, 6760, 7190, -2632, 413, -32, 1; 9: -1313480, 1922145, -1114740, 294370, -13580, -13251, 3976, -532, 36, -1;
%F FormulaThe row polynomials satisfy (2*n^3-74*n+26*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2-149*n-159-(n^3+3*n^2-89*n-91)*x-(17*n+17)*x^2+(n+1)*x^3)*P(n-1,x) - (21*n^2+21*n-546-(10*n^2+10*n-491)*x+(n^2+n-157)*x^2+21*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^3 + 15*n^2 - 149*n - 159 - (n^3 + 3*n^2 - 89*n - 91)*x - (17*n + 17)*x^2 + (n + 1)*x^3)*P[n - 1, x] - (21*n^2 + 21*n - 546 - (10*n^2 + 10*n - 491)*x + (n^2 + n - 157)*x^2 + 21*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 74*n + 26*n*x - 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^3 + 15*n^2 - 149*n - 159 - (n^3 + 3*n^2 - 89*n - 91)*x - (17*n + 17)*x^2 + (n + 1)*x^3)*v[i - 1] - (21*n^2 + 21*n - 546 - (10*n^2 + 10*n - 491)*x + (n^2 + n - 157)*x^2 + 21*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 74*n + 26*n*x - 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 - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 1
Proof chains ? JSON
References
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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
- [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