A110552 → basis A007318
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(4 + 3\,x - n - n x\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 2 | 5 | ||||||
| 2 | 10 | 13 | 4 | 27 | |||||
| 3 | 36 | 68 | 41 | 8 | 153 | ||||
| 4 | 136 | 332 | 294 | 113 | 16 | 891 | |||
| 5 | 528 | 1568 | 1816 | 1032 | 289 | 32 | 5265 | ||
| 6 | 2080 | 7248 | 10320 | 7720 | 3210 | 705 | 64 | 31347 | |
| 7 | 8256 | 32960 | 55536 | 51360 | 28220 | 9228 | 1665 | 128 | 187353 |
Compute more
?
Other recurrences
Image under the basis map A007318 · proved equal for all n (proof step)
\(P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
Technical details
- Node id
n-1ba7f2cb937015e6· revision 0 · JSON- Definition source
derived:shift_x_plus_onederived:basis_map_A007318(equal bye-ee6cba041ad56d92)- Fingerprint
f6-02204b3e41d44970
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): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 3, 2; 2: 10, 13, 4; 3: 36, 68, 41, 8; 4: 136, 332, 294, 113, 16; 5: 528, 1568, 1816, 1032, 289, 32; 6: 2080, 7248, 10320, 7720, 3210, 705, 64; 7: 8256, 32960, 55536, 51360, 28220, 9228, 1665, 128; 8: 32896, 147904, 287392, 315952, 215320, 93268, 25102, 3841, 256; 9: 131328, 656384, 1443584, 1836800, 1492064, 803264, 286832, 65552, 8705, 512;
%F FormulaThe row polynomials satisfy P(n,x) = -(n-4+(n-3)*x)*P(n-1,x) + (2+3*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[-(n - 4 + (n - 3)*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];
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] = -(n - 4 + (n - 3)*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], x)); 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
References
- [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