A119808 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-11 + 30\,x - 27\,x^{2} + 8\,x^{3} + n - 49\,n x^{2} + 24\,n x^{3} - 5\,n^{2} - n^{2} x + 24\,n^{2} x^{2} + 6\,n^{3} x\right)\,P_{n}(x) = \left(-22 + 60\,x - 54\,x^{2} + 16\,x^{3} - 14\,n + 7\,n x - 111\,n x^{2} - 4\,n x^{3} + 32\,n x^{4} - 9\,n^{2} + 13\,n^{2} x + 36\,n^{2} x^{2} + 32\,n^{2} x^{3} + 10\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-1 - 11\,x - 23\,x^{2} + 19\,x^{3} + 80\,x^{4} - 64\,x^{5} - n - 4\,n x + 37\,n x^{2} + 32\,n x^{3} - 64\,n x^{4} + 2\,n^{2} x + 14\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||
| 1 | 3 | 1 | 4 | |||
| 2 | 9 | 5 | 14 | |||
| 3 | 25 | 21 | 2 | 48 | ||
| 4 | 65 | 81 | 16 | 162 | ||
| 5 | 161 | 285 | 90 | 4 | 540 | |
| 6 | 385 | 921 | 432 | 44 | 1782 | |
| 7 | 897 | 2773 | 1842 | 312 | 8 | 5832 |
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Technical details
- Node id
n-b56cc1e9e3180cb8· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-a77d7304b4512bf6
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
- 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, 1; 2: 9, 5; 3: 25, 21, 2; 4: 65, 81, 16; 5: 161, 285, 90, 4; 6: 385, 921, 432, 44; 7: 897, 2773, 1842, 312, 8; 8: 2049, 7889, 7104, 1800, 112; 9: 4609, 21453, 25178, 9020, 960, 16;
%F FormulaThe row polynomials satisfy -(5*n^2-n+11-(6*n^3-n^2+30)*x-(24*n^2-49*n-27)*x^2-(24*n+8)*x^3)*P(n,x) = -(9*n^2+14*n+22-(10*n^3+13*n^2+7*n+60)*x-(8*n^3+36*n^2-111*n-54)*x^2-(32*n^2-4*n+16)*x^3-32*n*x^4)*P(n-1,x) - (n+1-(2*n^2-4*n-11)*x-(14*n^2+37*n-23)*x^2+(16*n^2-32*n-19)*x^3+(64*n-80)*x^4+64*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[(-(9*n^2 + 14*n + 22 - (10*n^3 + 13*n^2 + 7*n + 60)*x - (8*n^3 + 36*n^2 - 111*n - 54)*x^2 - (32*n^2 - 4*n + 16)*x^3 - 32*n*x^4)*P[n - 1, x] - (n + 1 - (2*n^2 - 4*n - 11)*x - (14*n^2 + 37*n - 23)*x^2 + (16*n^2 - 32*n - 19)*x^3 + (64*n - 80)*x^4 + 64*x^5)*D[P[n - 1, x], x])/(-5*n^2 + n - 11 + (6*n^3 - n^2 + 30)*x + (24*n^2 - 49*n - 27)*x^2 + (24*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] = (-(9*n^2 + 14*n + 22 - (10*n^3 + 13*n^2 + 7*n + 60)*x - (8*n^3 + 36*n^2 - 111*n - 54)*x^2 - (32*n^2 - 4*n + 16)*x^3 - 32*n*x^4)*v[i - 1] - (n + 1 - (2*n^2 - 4*n - 11)*x - (14*n^2 + 37*n - 23)*x^2 + (16*n^2 - 32*n - 19)*x^3 + (64*n - 80)*x^4 + 64*x^5)*deriv(v[i - 1], x))/(-5*n^2 + n - 11 + (6*n^3 - n^2 + 30)*x + (24*n^2 - 49*n - 27)*x^2 + (24*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.