A059344 → basis A013610 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-15 - 198\,x + 108\,x^{2} - 45\,n + 108\,n x + 27\,n^{2}\right)\,P_{n}(x) = \left(-3 + 180\,x + 432\,x^{2} - 648\,x^{3} + 33\,n - 216\,n x - 972\,n x^{2} + 648\,n x^{3} - 117\,n^{2} - 216\,n^{2} x + 648\,n^{2} x^{2} + 54\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 48\,x - 324\,x^{2} - 1080\,x^{3} - 1296\,x^{4} - 12\,n - 216\,n x - 972\,n x^{2} - 1296\,n x^{3} - 36\,n^{2} - 216\,n^{2} x - 324\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 9 | 6 | 15 | ||
| 3 | 25 | 18 | 43 | ||
| 4 | 133 | 324 | 108 | 565 | |
| 5 | 501 | 1500 | 540 | 2541 | |
| 6 | 2581 | 11970 | 14580 | 3240 | 32371 |
| 7 | 11509 | 63126 | 94500 | 22680 | 191815 |
Compute more
?
Technical details
- Node id
n-42ad00402123d391· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-4fd8c88feba28db3
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: 1; 2: 9, 6; 3: 25, 18; 4: 133, 324, 108; 5: 501, 1500, 540; 6: 2581, 11970, 14580, 3240; 7: 11509, 63126, 94500, 22680; 8: 61545, 433608, 1005480, 816480, 136080; 9: 308233, 2485944, 6817608, 6804000, 1224720;
%F FormulaThe row polynomials satisfy (27*n^2-45*n-15+(108*n-198)*x+108*x^2)*P(n,x) = (54*n^3-117*n^2+33*n-3+(162*n^3-216*n^2-216*n+180)*x+(648*n^2-972*n+432)*x^2+(648*n-648)*x^3)*P(n-1,x) - (36*n^2+12*n+4+(216*n^2+216*n+48)*x+(324*n^2+972*n+324)*x^2+(1296*n+1080)*x^3+1296*x^4)*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[((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*x^2)]];
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] = ((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*v[i - 1] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*x^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
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.