A119462 → basis A007318
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-2 - x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 2 | 2 | |||
| 2 | 4 | 2 | 6 | ||
| 3 | 8 | 6 | 14 | ||
| 4 | 16 | 16 | 2 | 34 | |
| 5 | 32 | 40 | 10 | 82 | |
| 6 | 64 | 96 | 36 | 2 | 198 |
| 7 | 128 | 224 | 112 | 14 | 478 |
Compute more
?
Other recurrences
Image under the basis map A007318 · proved equal for all n (proof step)
\(\left(-2 + n\right)\,P_{n}(x) = \left(-2\,x + n x\right)\,P_{n-2}(x) + \left(-4 + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 4 + 2\,x\)
Technical details
- Node id
n-cd85e79f8f3e8a20· revision 0 · JSON- Definition source
derived:shift_x_plus_onederived:basis_map_A007318(equal bye-f6823543a2580d4a)- Fingerprint
f6-702b8764f2c773f5
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2; 2: 4, 2; 3: 8, 6; 4: 16, 16, 2; 5: 32, 40, 10; 6: 64, 96, 36, 2; 7: 128, 224, 112, 14; 8: 256, 512, 320, 64, 2; 9: 512, 1152, 864, 240, 18;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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 ≥ 2
Proof chains ? JSON
Related ?
Same fingerprint ?
- A228955 → basis A007318
- A028326 → Veronese 2 → x ↦ x+1
- A003506 → Veronese 2 → x ↦ x+1
- A178820 → Veronese 2 → basis A007318
- A038208 → Veronese 2 → basis A007318
- basis-map coefficients of A113214, shifted → Veronese 2 → x ↦ x+1
- A178822 → Veronese 2 → x ↦ x+1
- A098158 → x ↦ x/(x+1)
- A178821 → Veronese 2 → x ↦ x+1
- A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).
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