A208509 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(24 - 32\,x - 6\,x^{2} + 4\,x^{3} - 50\,n + 81\,n x - 26\,n x^{2} + 4\,n x^{3} + 18\,n^{2} - 18\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} + n^{3} x\right)\,P_{n}(x) = \left(-12 + 70\,x - 132\,x^{2} + 46\,x^{3} - 4\,x^{4} - 26\,n + 21\,n x + 63\,n x^{2} - 26\,n x^{3} + 4\,n x^{4} + 6\,n^{2} + 8\,n^{2} x - 16\,n^{2} x^{2} + 4\,n^{2} x^{3} - 4\,n^{3} - 3\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-24\,x + 100\,x^{2} - 120\,x^{3} + 52\,x^{4} - 8\,x^{5} + 16\,n x - 50\,n x^{2} + 42\,n x^{3} - 8\,n x^{4} - 8\,n^{2} x + 10\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 3 | 3 | |||
| 2 | 6 | 1 | 7 | ||
| 3 | 12 | 5 | 17 | ||
| 4 | 23 | 16 | 1 | 40 | |
| 5 | 41 | 44 | 7 | 92 | |
| 6 | 68 | 109 | 30 | 1 | 208 |
| 7 | 106 | 245 | 104 | 9 | 464 |
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Technical details
- Node id
n-a3d8f451c3572045· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-fa1854099ac05d84
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: 3; 2: 6, 1; 3: 12, 5; 4: 23, 16, 1; 5: 41, 44, 7; 6: 68, 109, 30, 1; 7: 106, 245, 104, 9; 8: 157, 504, 314, 48, 1; 9: 223, 960, 846, 200, 11;
%F FormulaThe row polynomials satisfy -(4*n^3-18*n^2+50*n-24-(n^3-18*n^2+81*n-32)*x-(4*n^2-26*n-6)*x^2-(4*n+4)*x^3)*P(n,x) = -(4*n^3-6*n^2+26*n+12+(3*n^3-8*n^2-21*n-70)*x-(n^3-16*n^2+63*n-132)*x^2-(4*n^2-26*n+46)*x^3-(4*n-4)*x^4)*P(n-1,x) - ((8*n^2-16*n+24)*x-(10*n^2-50*n+100)*x^2+(2*n^2-42*n+120)*x^3+(8*n-52)*x^4+8*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[(-(4*n^3 - 6*n^2 + 26*n + 12 + (3*n^3 - 8*n^2 - 21*n - 70)*x - (n^3 - 16*n^2 + 63*n - 132)*x^2 - (4*n^2 - 26*n + 46)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] - ((8*n^2 - 16*n + 24)*x - (10*n^2 - 50*n + 100)*x^2 + (2*n^2 - 42*n + 120)*x^3 + (8*n - 52)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-4*n^3 + 18*n^2 - 50*n + 24 + (n^3 - 18*n^2 + 81*n - 32)*x + (4*n^2 - 26*n - 6)*x^2 + (4*n + 4)*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] = (-(4*n^3 - 6*n^2 + 26*n + 12 + (3*n^3 - 8*n^2 - 21*n - 70)*x - (n^3 - 16*n^2 + 63*n - 132)*x^2 - (4*n^2 - 26*n + 46)*x^3 - (4*n - 4)*x^4)*v[i - 1] - ((8*n^2 - 16*n + 24)*x - (10*n^2 - 50*n + 100)*x^2 + (2*n^2 - 42*n + 120)*x^3 + (8*n - 52)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-4*n^3 + 18*n^2 - 50*n + 24 + (n^3 - 18*n^2 + 81*n - 32)*x + (4*n^2 - 26*n - 6)*x^2 + (4*n + 4)*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.