A166341 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(n\right)\,P_{n}(x) = \left(-2 + 2\,x - 2\,n + n x^{2} - 2\,n^{2} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x + x^{2} - n x + n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 3\,n + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 10 + 2\,x\)
\(P_{2}(x) = 23 + 46\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 10 | 2 | 12 | |||||||
| 2 | 23 | 46 | 3 | 72 | ||||||
| 3 | 50 | 276 | 150 | 4 | 480 | |||||
| 4 | 105 | 1228 | 1842 | 420 | 5 | 3600 | ||||
| 5 | 216 | 4734 | 14736 | 9468 | 1080 | 6 | 30240 | |||
| 6 | 439 | 16794 | 94449 | 125932 | 41985 | 2634 | 7 | 282240 | ||
| 7 | 886 | 56528 | 528942 | 1259320 | 881570 | 169584 | 6202 | 8 | 2903040 | |
| 8 | 1781 | 183760 | 2709312 | 10528136 | 13160170 | 5418624 | 643160 | 14248 | 9 | 32659200 |
Compute more
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Technical details
- Node id
n-b1a3447bb8179bb6· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-0b93002f24124cd4
Observed on the stored rows ?
- Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 10, 2; 2: 23, 46, 3; 3: 50, 276, 150, 4; 4: 105, 1228, 1842, 420, 5; 5: 216, 4734, 14736, 9468, 1080, 6; 6: 439, 16794, 94449, 125932, 41985, 2634, 7; 7: 886, 56528, 528942, 1259320, 881570, 169584, 6202, 8; 8: 1781, 183760, 2709312, 10528136, 13160170, 5418624, 643160, 14248, 9; 9: 3572, 583338, 13042368, 77927592, 157922040, 116891388, 30432192, 2333352, 32148, 10; 10: 7155, 1821454, 60044535, 528756552, 1629398670, 1955278404, 925323966, 160118760, 8196543, 71550, 11;
%F FormulaThe row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 10 + 2*x, P(2,x) = 23 + 46*x + 3*x^2.
%t MathematicaP[1, x_] := 10 + 2*x;
P[2, x_] := 23 + 46*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 10 + 2*x; v[2] = 23 + 46*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
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.