A228955 → basis A007318
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(2\,n + n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||
| 1 | 2 | 1 | 3 | |||
| 2 | 8 | 6 | 14 | |||
| 3 | 48 | 48 | 6 | 102 | ||
| 4 | 384 | 480 | 120 | 984 | ||
| 5 | 3840 | 5760 | 2160 | 120 | 11880 | |
| 6 | 46080 | 80640 | 40320 | 5040 | 172080 | |
| 7 | 645120 | 1290240 | 806400 | 161280 | 5040 | 2908080 |
Compute more
?
Other recurrences
Image under the basis map A007318 · proved equal for all n (proof step)
\(P_{n}(x) = \left(-n x + n^{2} x\right)\,P_{n-2}(x) + \left(2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Technical details
- Node id
n-d7283fa0e9ac22b4· revision 0 · JSON- Definition source
derived:shift_x_plus_onederived:basis_map_A007318(equal bye-3e7135463492203d)- Fingerprint
f6-702b8764f2c773f5
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 8, 6; 3: 48, 48, 6; 4: 384, 480, 120; 5: 3840, 5760, 2160, 120; 6: 46080, 80640, 40320, 5040; 7: 645120, 1290240, 806400, 161280, 5040; 8: 10321920, 23224320, 17418240, 4838400, 362880; 9: 185794560, 464486400, 406425600, 145152000, 18144000, 362880;
%F FormulaThe row polynomials satisfy P(n,x) = (2*n+n*x)*P(n-1,x) - (2*x+2*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[(2*n + n*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + n*x)*v[i - 1] - (2*x + 2*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
Related ?
Same fingerprint ?
- A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).
- basis-map coefficients of A113214, shifted → Veronese 2 → x ↦ x+1
- A119462 → basis A007318
- A003506 → Veronese 2 → x ↦ x+1
- A178820 → Veronese 2 → basis A007318
- A038208 → Veronese 2 → basis A007318
- A178821 → Veronese 2 → x ↦ x+1
- A028326 → Veronese 2 → x ↦ x+1
- A178822 → Veronese 2 → x ↦ x+1
- A098158 → x ↦ x/(x+1)
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