A108299 → D+1
Proved for all n: all roots real what this means
Recurrence
\(\left(-75 + 72\,x - 12\,x^{2} + 110\,n - 72\,n x + 8\,n x^{2} - 52\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-36 + 39\,x + 112\,x^{2} - 132\,x^{3} + 32\,x^{4} + 66\,n - 44\,n x - 180\,n x^{2} + 128\,n x^{3} - 16\,n x^{4} - 40\,n^{2} + 12\,n^{2} x + 96\,n^{2} x^{2} - 32\,n^{2} x^{3} + 8\,n^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-9\,x + 12\,x^{2} + 32\,x^{3} - 48\,x^{4} + 16\,x^{5} + 12\,n x - 8\,n x^{2} - 48\,n x^{3} + 32\,n x^{4} - 4\,n^{2} x + 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
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 | 0 | -3 | -1 | -4 | ||||||||||||
| 4 | 0 | -5 | 1 | 1 | -3 | |||||||||||
| 5 | 0 | -7 | 3 | 6 | 1 | 3 | ||||||||||
| 6 | 0 | -9 | 5 | 15 | -2 | -1 | 8 | |||||||||
| 7 | 0 | -11 | 7 | 28 | -9 | -9 | -1 | 5 | ||||||||
| 8 | 0 | -13 | 9 | 45 | -20 | -30 | 3 | 1 | -5 | |||||||
| 9 | 0 | -15 | 11 | 66 | -35 | -70 | 18 | 12 | 1 | -12 | ||||||
| 10 | 0 | -17 | 13 | 91 | -54 | -135 | 50 | 50 | -4 | -1 | -7 | |||||
| 11 | 0 | -19 | 15 | 120 | -77 | -231 | 105 | 140 | -30 | -15 | -1 | 7 | ||||
| 12 | 0 | -21 | 17 | 153 | -104 | -364 | 189 | 315 | -100 | -75 | 5 | 1 | 16 | |||
| 13 | 0 | -23 | 19 | 190 | -135 | -540 | 308 | 616 | -245 | -245 | 45 | 18 | 1 | 9 | ||
| 14 | 0 | -25 | 21 | 231 | -170 | -765 | 468 | 1092 | -504 | -630 | 175 | 105 | -6 | -1 | -9 | |
| 15 | 0 | -27 | 23 | 276 | -209 | -1045 | 675 | 1800 | -924 | -1386 | 490 | 392 | -63 | -21 | -1 | -20 |
Computed in your browser ?
Technical details
- Node id
n-af9192a4a07453ac· revision 1 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-b9b953b2e3c32be8
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
OEIS matches ?
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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 0, -1; 3: 0, -3, -1; 4: 0, -5, 1, 1; 5: 0, -7, 3, 6, 1; 6: 0, -9, 5, 15, -2, -1; 7: 0, -11, 7, 28, -9, -9, -1; 8: 0, -13, 9, 45, -20, -30, 3, 1; 9: 0, -15, 11, 66, -35, -70, 18, 12, 1; 10: 0, -17, 13, 91, -54, -135, 50, 50, -4, -1;
%F FormulaThe row polynomials satisfy (8*n^3-52*n^2+110*n-75+(16*n^2-72*n+72)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-40*n^2+66*n-36+(12*n^2-44*n+39)*x-(16*n^3-96*n^2+180*n-112)*x^2-(32*n^2-128*n+132)*x^3-(16*n-32)*x^4)*P(n-1,x) - ((4*n^2-12*n+9)*x+(8*n-12)*x^2-(16*n^2-48*n+32)*x^3-(32*n-48)*x^4-16*x^5)*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[((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*P[n - 1, x] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*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] = ((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*v[i - 1] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*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)Generated with polytool c48629374597.
Proved properties
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Real-rooted. Every row P_n, n >= 1, 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-b113ea8f5eac0e8f
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
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preserver_image derivative_polynomial, shift 0
e-b113ea8f5eac0e8fLaguerre’s theorem on differential operators [SP14] polytool ?
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premise (sequence):
Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
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theorem_check three_term_recurrence
e-7467667c9a732a94root: theorem checkThree-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
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premise (sequence):
Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
Related sequences ?
No candidate equalities or related sequences.
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