A129710 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(6 - 24\,x - 10\,x^{2} + 8\,x^{3} - 13\,n + 31\,n x - 9\,n x^{2} + 4\,n x^{3} + 9\,n^{2} - 8\,n^{2} x + 4\,n^{2} x^{2} - 2\,n^{3} + n^{3} x\right)\,P_{n}(x) = \left(2 - 11\,x - 73\,x^{2} + 46\,x^{3} - 8\,x^{4} - 7\,n + 59\,n x^{2} - 30\,n x^{3} + 8\,n x^{4} + 7\,n^{2} + 14\,n^{2} x - 18\,n^{2} x^{2} + 8\,n^{2} x^{3} - 2\,n^{3} - 3\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x - 6\,x^{2} - 31\,x^{3} + 32\,x^{4} - 16\,x^{5} - 3\,n x - 4\,n x^{2} + 28\,n x^{3} - 16\,n x^{4} + 2\,n^{2} x + 7\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 2 | 2 | |||
| 2 | 3 | 1 | 4 | ||
| 3 | 5 | 3 | 8 | ||
| 4 | 7 | 7 | 1 | 15 | |
| 5 | 9 | 15 | 4 | 28 | |
| 6 | 11 | 27 | 12 | 1 | 51 |
| 7 | 13 | 43 | 31 | 5 | 92 |
Compute more
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Technical details
- Node id
n-01db3e9311c87670· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-85749c986becf742
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: 3, 1; 3: 5, 3; 4: 7, 7, 1; 5: 9, 15, 4; 6: 11, 27, 12, 1; 7: 13, 43, 31, 5; 8: 15, 63, 67, 18, 1; 9: 17, 87, 126, 54, 6;
%F FormulaThe row polynomials satisfy -(2*n^3-9*n^2+13*n-6-(n^3-8*n^2+31*n-24)*x-(4*n^2-9*n-10)*x^2-(4*n+8)*x^3)*P(n,x) = -(2*n^3-7*n^2+7*n-2+(3*n^3-14*n^2+11)*x-(2*n^3-18*n^2+59*n-73)*x^2-(8*n^2-30*n+46)*x^3-(8*n-8)*x^4)*P(n-1,x) + ((2*n^2-3*n+1)*x+(7*n^2-4*n-6)*x^2-(4*n^2-28*n+31)*x^3-(16*n-32)*x^4-16*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[(-(2*n^3 - 7*n^2 + 7*n - 2 + (3*n^3 - 14*n^2 + 11)*x - (2*n^3 - 18*n^2 + 59*n - 73)*x^2 - (8*n^2 - 30*n + 46)*x^3 - (8*n - 8)*x^4)*P[n - 1, x] + ((2*n^2 - 3*n + 1)*x + (7*n^2 - 4*n - 6)*x^2 - (4*n^2 - 28*n + 31)*x^3 - (16*n - 32)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 9*n^2 - 13*n + 6 + (n^3 - 8*n^2 + 31*n - 24)*x + (4*n^2 - 9*n - 10)*x^2 + (4*n + 8)*x^3)]];
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^3 - 7*n^2 + 7*n - 2 + (3*n^3 - 14*n^2 + 11)*x - (2*n^3 - 18*n^2 + 59*n - 73)*x^2 - (8*n^2 - 30*n + 46)*x^3 - (8*n - 8)*x^4)*v[i - 1] + ((2*n^2 - 3*n + 1)*x + (7*n^2 - 4*n - 6)*x^2 - (4*n^2 - 28*n + 31)*x^3 - (16*n - 32)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(-2*n^3 + 9*n^2 - 13*n + 6 + (n^3 - 8*n^2 + 31*n - 24)*x + (4*n^2 - 9*n - 10)*x^2 + (4*n + 8)*x^3)); 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 (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.