A179900 → x ↦ x+1
Proved for all n all roots real, in (2, 6)alternating coefficient signsinterlacing (weak)
Recurrence
\(\left(2\,n\right)\,P_{n}(x) = \left(4 - x + 4\,n - n x\right)\,P_{n-1}(x) + \left(-12 + 8\,x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 4 | -1 | 3 | |||||||||||||
| 2 | 15 | -8 | 1 | 8 | ||||||||||||
| 3 | 56 | -46 | 12 | -1 | 21 | |||||||||||
| 4 | 209 | -232 | 93 | -16 | 1 | 55 | ||||||||||
| 5 | 780 | -1091 | 592 | -156 | 20 | -1 | 144 | |||||||||
| 6 | 2911 | -4912 | 3366 | -1200 | 235 | -24 | 1 | 377 | ||||||||
| 7 | 10864 | -21468 | 17784 | -8010 | 2120 | -330 | 28 | -1 | 987 | |||||||
| 8 | 40545 | -91824 | 89238 | -48624 | 16255 | -3416 | 441 | -32 | 1 | 2584 | ||||||
| 9 | 151316 | -386373 | 430992 | -275724 | 111524 | -29589 | 5152 | -568 | 36 | -1 | 6765 | |||||
| 10 | 564719 | -1604984 | 2021103 | -1485264 | 705565 | -226464 | 49756 | -7392 | 711 | -40 | 1 | 17711 | ||||
| 11 | 2107560 | -6598282 | 9258404 | -7686435 | 4196000 | -1581832 | 420336 | -78756 | 10200 | -870 | 44 | -1 | 46368 | |||
| 12 | 7865521 | -26895704 | 41610795 | -38518880 | 23764870 | -10296864 | 3213420 | -727968 | 118845 | -13640 | 1045 | -48 | 1 | 121393 | ||
| 13 | 29354524 | -108850055 | 184080480 | -187999880 | 129382360 | -63370494 | 22730208 | -6046536 | 1193148 | -172535 | 17776 | -1236 | 52 | -1 | 317811 | |
| 14 | 109552575 | -437859040 | 803561180 | -897561120 | 681764450 | -372567472 | 151077906 | -46188384 | 10700283 | -1869648 | 242594 | -22672 | 1443 | -56 | 1 | 832040 |
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Technical details
- Node id
n-a9419a0f5f0cae19· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-000cdfc9e19dff85
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: 15, -8, 1; 3: 56, -46, 12, -1; 4: 209, -232, 93, -16, 1; 5: 780, -1091, 592, -156, 20, -1; 6: 2911, -4912, 3366, -1200, 235, -24, 1; 7: 10864, -21468, 17784, -8010, 2120, -330, 28, -1; 8: 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1; 9: 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1;
%F FormulaThe row polynomials satisfy 2*n*P(n,x) = (4*n+4-(n+1)*x)*P(n-1,x) - (12-8*x+x^2)*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[((4*n + 4 - (n + 1)*x)*P[n - 1, x] - (12 - 8*x + x^2)*D[P[n - 1, x], x])/(2*n)]];
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] = ((4*n + 4 - (n + 1)*x)*v[i - 1] - (12 - 8*x + x^2)*deriv(v[i - 1], x))/(2*n)); 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 (2, 6)
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n(-x) have one sign
- 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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023