A208509 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(16 - x - 9\,n + 5\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(24 + 40\,x - 2\,x^{2} - 12\,n - 14\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(12\,x + 12\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 3 | 3 | |||
| 2 | 6 | 1 | 7 | ||
| 3 | 12 | 5 | 17 | ||
| 4 | 24 | 16 | 1 | 41 | |
| 5 | 48 | 44 | 7 | 99 | |
| 6 | 96 | 112 | 30 | 1 | 239 |
| 7 | 192 | 272 | 104 | 9 | 577 |
Compute more
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Technical details
- Node id
n-c6da825af22846f3· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-3e306556aac30f2d
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: 3; 2: 6, 1; 3: 12, 5; 4: 24, 16, 1; 5: 48, 44, 7; 6: 96, 112, 30, 1; 7: 192, 272, 104, 9; 8: 384, 640, 320, 48, 1; 9: 768, 1472, 912, 200, 11;
%F FormulaThe row polynomials satisfy (n^2+2*n+1)*P(n,x) = (5*n^2-9*n+16+(n^2-1)*x)*P(n-1,x) - (12*n-24+(14*n-40)*x+(2*n+2)*x^2)*d/dx P(n-1,x) + (12*x+12*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*P[n - 1, x] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x] + (12*x + 12*x^2)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n + 1)]];
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] = ((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*v[i - 1] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x) + (12*x + 12*x^2)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*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)
Proof chains ? JSON
Related ?
Candidate equalities ?
- A208509 → basis A007318 15 rows; no all-index equality certificate was found
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