A144698 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 11\,x^{2} + 5\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 5 | 6 | ||||||
| 2 | 1 | 15 | 30 | 46 | |||||
| 3 | 1 | 35 | 180 | 210 | 426 | ||||
| 4 | 1 | 75 | 750 | 2100 | 1680 | 4606 | |||
| 5 | 1 | 155 | 2700 | 13650 | 25200 | 15120 | 56826 | ||
| 6 | 1 | 315 | 9030 | 73500 | 235200 | 317520 | 151200 | 786766 | |
| 7 | 1 | 635 | 28980 | 357210 | 1764000 | 4021920 | 4233600 | 1663200 | 12069546 |
Compute more
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Technical details
- Node id
n-51dc433de0f0e26a· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-2a9069f1f4217ff4
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 5; 2: 1, 15, 30; 3: 1, 35, 180, 210; 4: 1, 75, 750, 2100, 1680; 5: 1, 155, 2700, 13650, 25200, 15120; 6: 1, 315, 9030, 73500, 235200, 317520, 151200; 7: 1, 635, 28980, 357210, 1764000, 4021920, 4233600, 1663200; 8: 1, 1275, 90750, 1631700, 11677680, 40007520, 69854400, 59875200, 19958400; 9: 1, 2555, 279900, 7162050, 71442000, 345144240, 889056000, 1247400000, 898128000, 259459200;
%F FormulaThe row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+7*x+11*x^2+5*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[Together[((1 + 7*x + 11*x^2 + 5*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((1 + 7*x + 11*x^2 + 5*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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, 0]
- 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
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