A060187 → γ → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-3 + n\right)\,P_{n}(x) = \left(6 - 80\,x + 480\,x^{2} - 3\,n + 56\,n x - 496\,n x^{2} - 8\,n^{2} x + 160\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 64\,x^{2} + 192\,x^{3} - 2\,n x + 48\,n x^{2} - 160\,n x^{3} - 8\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-11 + 96\,x + 4\,n - 56\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 24\,x^{2} + 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | Pn(1) |
|---|---|---|---|---|
| 1 | 0 | |||
| 2 | 0 | |||
| 3 | 4 | 4 | ||
| 4 | 20 | 20 | ||
| 5 | 72 | 160 | 232 | |
| 6 | 232 | 1952 | 2184 | |
| 7 | 716 | 15328 | 11712 | 27756 |
| 8 | 2172 | 99552 | 265920 | 367644 |
Compute more
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Technical details
- Node id
n-18bae3f267a0e827· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-f9921f80f8a8f61e
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 -3)
- 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: 0; 2: 0; 3: 4; 4: 20; 5: 72, 160; 6: 232, 1952; 7: 716, 15328, 11712; 8: 2172, 99552, 265920; 9: 6544, 584640, 3653376, 1418240; 10: 19664, 3237312, 39604992, 51733504;
%F FormulaThe row polynomials satisfy (n-3)*P(n,x) = -(3*n-6+(8*n^2-56*n+80)*x+(16*n^3-160*n^2+496*n-480)*x^2)*P(n-2,x) - ((2*n-4)*x+(8*n^2-48*n+64)*x^2-(32*n^2-160*n+192)*x^3)*d/dx P(n-2,x) + (4*n-11+(8*n^2-56*n+96)*x)*P(n-1,x) + ((2*n-6)*x-(8*n-24)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 4.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (8*n^2 - 56*n + 80)*x + (16*n^3 - 160*n^2 + 496*n - 480)*x^2)*P[n - 2, x] - ((2*n - 4)*x + (8*n^2 - 48*n + 64)*x^2 - (32*n^2 - 160*n + 192)*x^3)*D[P[n - 2, x], x] + (4*n - 11 + (8*n^2 - 56*n + 96)*x)*P[n - 1, x] + ((2*n - 6)*x - (8*n - 24)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 4, N, my(i = n); v[i] = (-(3*n - 6 + (8*n^2 - 56*n + 80)*x + (16*n^3 - 160*n^2 + 496*n - 480)*x^2)*v[i - 2] - ((2*n - 4)*x + (8*n^2 - 48*n + 64)*x^2 - (32*n^2 - 160*n + 192)*x^3)*deriv(v[i - 2], x) + (4*n - 11 + (8*n^2 - 56*n + 96)*x)*v[i - 1] + ((2*n - 6)*x - (8*n - 24)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) =…?
Related ?
Same fingerprint ?
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.