A034807 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-2\,n - 6\,n x + n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | Pn(1) |
|---|---|---|---|---|
| 0 | 0 | |||
| 1 | 0 | |||
| 2 | 2 | 2 | ||
| 3 | 3 | 3 | ||
| 4 | 4 | 4 | 8 | |
| 5 | 5 | 10 | 15 | |
| 6 | 6 | 18 | 6 | 30 |
| 7 | 7 | 28 | 21 | 56 |
Compute more
?
Technical details
- Node id
n-fbc0ddcc0e322ee5· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-8de4b75c4559d58b
Observed on the stored rows ?
- Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
- 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: 0; 2: 2; 3: 3; 4: 4, 4; 5: 5, 10; 6: 6, 18, 6; 7: 7, 28, 21; 8: 8, 40, 48, 8; 9: 9, 54, 90, 36;
%F FormulaThe row polynomials satisfy (n^2-3*n+2)*P(n,x) = (n^2-2*n+(2*n^2-6*n)*x)*P(n-1,x) - (n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 2*n + (2*n^2 - 6*n)*x)*P[n - 1, x] - (n*x + 4*n*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 - 2*n + (2*n^2 - 6*n)*x)*v[i - 1] - (n*x + 4*n*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 2)); 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 ?
- A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).
- A178820 → Veronese 2 (j=1) → basis A013611
- reversed rows → D
- A113214 → θ+1
- A178822 → Veronese 2 (j=1) → basis A013611
- A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
- A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of …
- A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
- basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → basis A013611
- A178821 → Veronese 2 (j=1) → basis A013611
- A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
- A038208 → Veronese 2 (j=1) → basis A013611
- A028326 → Veronese 2 (j=1) → basis A013611
- A034867 → basis A013611
- A007318 → Veronese 2 (j=1) → basis A013611
- A113214 → θ+1 → reverse
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.