A142075 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(2 - x + n x\right)\,P_{n-1}(x) + \left(2\,x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 1 | 3 | ||||||
| 3 | 4 | 8 | 1 | 13 | |||||
| 4 | 8 | 44 | 22 | 1 | 75 | ||||
| 5 | 16 | 208 | 264 | 52 | 1 | 541 | |||
| 6 | 32 | 912 | 2416 | 1208 | 114 | 1 | 4683 | ||
| 7 | 64 | 3840 | 19056 | 19328 | 4764 | 240 | 1 | 47293 | |
| 8 | 128 | 15808 | 137376 | 249904 | 124952 | 17172 | 494 | 1 | 545835 |
Compute more
?
Technical details
- Node id
n-ce7e2a3941a4e8f5· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-735f72d4be30f803
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): yes - 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: 2, 1; 3: 4, 8, 1; 4: 8, 44, 22, 1; 5: 16, 208, 264, 52, 1; 6: 32, 912, 2416, 1208, 114, 1; 7: 64, 3840, 19056, 19328, 4764, 240, 1; 8: 128, 15808, 137376, 249904, 124952, 17172, 494, 1; 9: 256, 64256, 934912, 2823488, 2499040, 705872, 58432, 1004, 1; 10: 512, 259328, 6123520, 29132288, 41931328, 20965664, 3641536, 191360, 2026, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (2+(n-1)*x)*P(n-1,x) + (2*x-x^2)*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[(2 + (n - 1)*x)*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
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] = (2 + (n - 1)*x)*v[i - 1] + (2*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, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
- A156365 → reverse
- A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
- A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally…
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