A152828 → basis A007318 → D
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-60 + 47\,n - 12\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(432 + 216\,x - 104\,n - 52\,n x - 6\,n^{2} - 3\,n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-3}(x) + \left(756 + 756\,x + 189\,x^{2} - 108\,n - 108\,n x - 27\,n x^{2}\right)\,P_{n-3}'(x) + \left(216 + 324\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-3}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)
\(P_{4}(x) = 1\)
\(P_{5}(x) = 1\)
Rows ?
| n | x0 | x1 | Pn(1) |
|---|---|---|---|
| 0 | 0 | ||
| 1 | 0 | ||
| 2 | 0 | ||
| 3 | 1 | 1 | |
| 4 | 1 | 1 | |
| 5 | 1 | 1 | |
| 6 | 4 | 2 | 6 |
| 7 | 4 | 2 | 6 |
Compute more
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Technical details
- Node id
n-7bc0dafb54ade7ab· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-19c649f255ec1907
Observed on the stored rows ?
- Degrees: -1, -1, -1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3 (irregular)
- 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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 0; 2: 0; 3: 1; 4: 1; 5: 1; 6: 4, 2; 7: 4, 2; 8: 4, 2; 9: 12, 12, 3;
%F FormulaThe row polynomials satisfy (n^3-12*n^2+47*n-60)*P(n,x) = (2*n^3-6*n^2-104*n+432+(n^3-3*n^2-52*n+216)*x)*P(n-3,x) - (108*n-756+(108*n-756)*x+(27*n-189)*x^2)*d/dx P(n-3,x) + (216+324*x+162*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1, P(4,x) = 1, P(5,x) = 1.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[4, x_] := 1;
P[5, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 6*n^2 - 104*n + 432 + (n^3 - 3*n^2 - 52*n + 216)*x)*P[n - 3, x] - (108*n - 756 + (108*n - 756)*x + (27*n - 189)*x^2)*D[P[n - 3, x], x] + (216 + 324*x + 162*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 - 12*n^2 + 47*n - 60)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 1; v[5] = 1; v[6] = 1; for(n = 6, N, my(i = n + 1); v[i] = ((2*n^3 - 6*n^2 - 104*n + 432 + (n^3 - 3*n^2 - 52*n + 216)*x)*v[i - 3] - (108*n - 756 + (108*n - 756)*x + (27*n - 189)*x^2)*deriv(v[i - 3], x) + (216 + 324*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 - 12*n^2 + 47*n - 60)); 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
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.