A131222 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-288 + 408\,n - 212\,n^{2} + 48\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-180 + 24\,x + 222\,n - 20\,n x - 90\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-52 + 12\,x - x^{2} + 52\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)
\(P_{3}(x) = 14 + 18\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 3 | 2 | 5 | |||||
| 3 | 14 | 18 | 3 | 35 | ||||
| 4 | 90 | 166 | 54 | 4 | 314 | |||
| 5 | 744 | 1740 | 825 | 120 | 5 | 3434 | ||
| 6 | 7560 | 20948 | 12825 | 2740 | 225 | 6 | 44304 | |
| 7 | 91440 | 287784 | 212772 | 59220 | 7175 | 378 | 7 | 658776 |
Compute more
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Technical details
- Node id
n-890b2f4d886ce308· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-3b33a02937f0bff7
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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: 0; 1: 1; 2: 3, 2; 3: 14, 18, 3; 4: 90, 166, 54, 4; 5: 744, 1740, 825, 120, 5; 6: 7560, 20948, 12825, 2740, 225, 6; 7: 91440, 287784, 212772, 59220, 7175, 378, 7; 8: 1285200, 4466712, 3822588, 1297156, 205800, 16044, 588, 8; 9: 20603520, 77539680, 74611716, 29595888, 5757045, 589680, 32046, 864, 9;
%F FormulaThe row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(4*n^2-20*n+24)*x)*P(n-3,x) - (13*n^2-52*n+52+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 14 + 18*x + 3*x^2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 14 + 18*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*P[n - 3, x] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3 + 2*x; v[4] = 14 + 18*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*v[i - 3] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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
Related ?
Same fingerprint ?
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.