A181371 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-6\,n - 27\,n x + 12\,n x^{2} + 6\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(2 - 12\,x + 18\,x^{2} - 8\,x^{3} - 25\,n - 65\,n x + 10\,n x^{2} + 8\,n x^{3} + 25\,n^{2} + 30\,n^{2} x + 8\,n^{2} x^{2} + 7\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-15 - 2\,x + 33\,x^{2} - 16\,x^{4} + 20\,n + 16\,n x - 20\,n x^{2} - 16\,n x^{3} + 5\,n^{2} - n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 3 | |||||||
| 2 | 9 | 1 | 10 | ||||||
| 3 | 27 | 6 | 33 | ||||||
| 4 | 80 | 27 | 1 | 108 | |||||
| 5 | 234 | 108 | 9 | 351 | |||||
| 6 | 677 | 402 | 54 | 1 | 1134 | ||||
| 7 | 1941 | 1422 | 270 | 12 | 3645 | ||||
| 8 | 5523 | 4841 | 1209 | 90 | 1 | 11664 | |||
| 9 | 15615 | 15996 | 5013 | 540 | 15 | 37179 | |||
| 10 | 43906 | 51609 | 19622 | 2825 | 135 | 1 | 118098 | ||
| 11 | 122868 | 163296 | 73422 | 13428 | 945 | 18 | 373977 | ||
| 12 | 342409 | 508380 | 264960 | 59386 | 5655 | 189 | 1 | 1180980 | |
| 13 | 950727 | 1561212 | 928152 | 248160 | 30303 | 1512 | 21 | 3720087 | |
| 14 | 2631165 | 4738681 | 3171558 | 990480 | 149380 | 10185 | 252 | 1 | 11691702 |
Compute more
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Technical details
- Node id
n-5115618c5fe87f9f· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-103c5766d3e94b1f
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: 9, 1; 3: 27, 6; 4: 80, 27, 1; 5: 234, 108, 9; 6: 677, 402, 54, 1; 7: 1941, 1422, 270, 12; 8: 5523, 4841, 1209, 90, 1; 9: 15615, 15996, 5013, 540, 15;
%F FormulaThe row polynomials satisfy (3*n^3+6*n^2-6*n+(12*n^2-27*n)*x+12*n*x^2)*P(n,x) = (7*n^3+25*n^2-25*n+2+(2*n^3+30*n^2-65*n-12)*x+(8*n^2+10*n+18)*x^2+(8*n-8)*x^3)*P(n-1,x) + (5*n^2+20*n-15-(n^2-16*n+2)*x-(4*n^2+20*n-33)*x^2-16*n*x^3-16*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[((7*n^3 + 25*n^2 - 25*n + 2 + (2*n^3 + 30*n^2 - 65*n - 12)*x + (8*n^2 + 10*n + 18)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + (5*n^2 + 20*n - 15 - (n^2 - 16*n + 2)*x - (4*n^2 + 20*n - 33)*x^2 - 16*n*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n^3 + 6*n^2 - 6*n + (12*n^2 - 27*n)*x + 12*n*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] = ((7*n^3 + 25*n^2 - 25*n + 2 + (2*n^3 + 30*n^2 - 65*n - 12)*x + (8*n^2 + 10*n + 18)*x^2 + (8*n - 8)*x^3)*v[i - 1] + (5*n^2 + 20*n - 15 - (n^2 - 16*n + 2)*x - (4*n^2 + 20*n - 33)*x^2 - 16*n*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n^3 + 6*n^2 - 6*n + (12*n^2 - 27*n)*x + 12*n*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.