A115951 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-4\,n x - 6\,n x^{2} + 4\,n x^{3} - n^{2} + 2\,n^{2} x + 4\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 16\,x^{2} + 4\,x^{3} + 8\,x^{4} - 6\,n + 2\,n x + 16\,n x^{2} + 8\,n^{2} + 6\,n^{2} x - 8\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 24\,x^{4} + 16\,x^{5} - 4\,n x + 4\,n x^{2} + 24\,n x^{3} + 16\,n x^{4} + 4\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 2 | 4 | ||||||
| 2 | 2 | 14 | 6 | 22 | |||||
| 3 | 0 | 24 | 72 | 20 | 116 | ||||
| 4 | 0 | 12 | 186 | 340 | 70 | 608 | |||
| 5 | 0 | 0 | 180 | 1180 | 1540 | 252 | 3152 | ||
| 6 | 0 | 0 | 60 | 1700 | 6720 | 6804 | 924 | 16208 | |
| 7 | 0 | 0 | 0 | 1120 | 12880 | 35784 | 29568 | 3432 | 82784 |
Compute more
?
Technical details
- Node id
n-27decca1d071fa2c· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-8175414b3da05124
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: 2, 2; 2: 2, 14, 6; 3: 0, 24, 72, 20; 4: 0, 12, 186, 340, 70; 5: 0, 0, 180, 1180, 1540, 252; 6: 0, 0, 60, 1700, 6720, 6804, 924; 7: 0, 0, 0, 1120, 12880, 35784, 29568, 3432; 8: 0, 0, 0, 280, 12670, 85680, 182028, 126984, 12870; 9: 0, 0, 0, 0, 6300, 112140, 522984, 895752, 540540, 48620;
%F FormulaThe row polynomials satisfy (n^3-n^2+(2*n^2-4*n)*x+(4*n^2-6*n)*x^2+4*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(6*n^2+2*n-12)*x+(8*n^2-16*n+16)*x^2-4*x^3-8*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(4*n^2+4*n-8)*x^2+24*n*x^3+(16*n+24)*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 - 8*n^2 + 6*n - (6*n^2 + 2*n - 12)*x + (8*n^2 - 16*n + 16)*x^2 - 4*x^3 - 8*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (4*n^2 + 4*n - 8)*x^2 + 24*n*x^3 + (16*n + 24)*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (2*n^2 - 4*n)*x + (4*n^2 - 6*n)*x^2 + 4*n*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 - 8*n^2 + 6*n - (6*n^2 + 2*n - 12)*x + (8*n^2 - 16*n + 16)*x^2 - 4*x^3 - 8*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (4*n^2 + 4*n - 8)*x^2 + 24*n*x^3 + (16*n + 24)*x^4 + 16*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 + (2*n^2 - 4*n)*x + (4*n^2 - 6*n)*x^2 + 4*n*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]
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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.