A110552 → x ↦ x/(x+1)
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 13\,x^{2} + 9\,x^{3} + 2\,x^{4} - 2\,n x - 6\,n x^{2} - 6\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 3 | 4 | ||||||
| 2 | 1 | 7 | 10 | 18 | |||||
| 3 | 1 | 13 | 40 | 36 | 90 | ||||
| 4 | 1 | 23 | 114 | 212 | 136 | 486 | |||
| 5 | 1 | 41 | 288 | 824 | 1072 | 528 | 2754 | ||
| 6 | 1 | 75 | 690 | 2680 | 5280 | 5232 | 2080 | 16038 | |
| 7 | 1 | 141 | 1608 | 7900 | 20880 | 31152 | 24832 | 8256 | 94770 |
Compute more
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Technical details
- Node id
n-5f3f02de594824b0· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-13f9c16bd5f73dee
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 3; 2: 1, 7, 10; 3: 1, 13, 40, 36; 4: 1, 23, 114, 212, 136; 5: 1, 41, 288, 824, 1072, 528; 6: 1, 75, 690, 2680, 5280, 5232, 2080; 7: 1, 141, 1608, 7900, 20880, 31152, 24832, 8256; 8: 1, 271, 3682, 21868, 72520, 144592, 173152, 115264, 32896; 9: 1, 529, 8320, 57904, 230944, 576352, 921088, 920320, 525568, 131328;
%F FormulaThe row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(2*n-7)*x-(6*n-13)*x^2-(6*n-9)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*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 - (2*n - 7)*x - (6*n - 13)*x^2 - (6*n - 9)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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 - (2*n - 7)*x - (6*n - 13)*x^2 - (6*n - 9)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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