A209599 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-20 - 20\,x + 14\,x^{2} + 8\,x^{3} + 13\,n + 25\,n x + 3\,n x^{2} + 4\,n x^{3} + 5\,n^{2} + 4\,n^{2} x^{2} - n^{3} + n^{3} x\right)\,P_{n}(x) = \left(-44 - 51\,x + 17\,x^{2} + 14\,x^{3} - 8\,x^{4} + 30\,n + 60\,n x + 17\,n x^{2} + 2\,n x^{3} + 8\,n x^{4} + 11\,n^{2} + 2\,n^{2} x + 6\,n^{2} x^{2} + 8\,n^{2} x^{3} - 3\,n^{3} + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-20 - 56\,x - 67\,x^{2} - 63\,x^{3} - 48\,x^{4} - 16\,x^{5} + 5\,n + 9\,n x - 16\,n x^{2} - 36\,n x^{3} - 16\,n x^{4} + 5\,n^{2} + 4\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 2 | 2 | |||
| 2 | 4 | 1 | 5 | ||
| 3 | 8 | 3 | 11 | ||
| 4 | 15 | 9 | 1 | 25 | |
| 5 | 28 | 23 | 4 | 55 | |
| 6 | 51 | 54 | 15 | 1 | 121 |
| 7 | 92 | 120 | 46 | 5 | 263 |
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Technical details
- Node id
n-708cc2529cf30076· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-f5d275958b251de1
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: 2; 2: 4, 1; 3: 8, 3; 4: 15, 9, 1; 5: 28, 23, 4; 6: 51, 54, 15, 1; 7: 92, 120, 46, 5; 8: 164, 255, 127, 22, 1; 9: 290, 525, 324, 78, 6;
%F FormulaThe row polynomials satisfy -(n^3-5*n^2-13*n+20-(n^3+25*n-20)*x-(4*n^2+3*n+14)*x^2-(4*n+8)*x^3)*P(n,x) = -(3*n^3-11*n^2-30*n+44-(n^3+2*n^2+60*n-51)*x-(2*n^3+6*n^2+17*n+17)*x^2-(8*n^2+2*n+14)*x^3-(8*n-8)*x^4)*P(n-1,x) + (5*n^2+5*n-20+(4*n^2+9*n-56)*x-(5*n^2+16*n+67)*x^2-(4*n^2+36*n+63)*x^3-(16*n+48)*x^4-16*x^5)*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[(-(3*n^3 - 11*n^2 - 30*n + 44 - (n^3 + 2*n^2 + 60*n - 51)*x - (2*n^3 + 6*n^2 + 17*n + 17)*x^2 - (8*n^2 + 2*n + 14)*x^3 - (8*n - 8)*x^4)*P[n - 1, x] + (5*n^2 + 5*n - 20 + (4*n^2 + 9*n - 56)*x - (5*n^2 + 16*n + 67)*x^2 - (4*n^2 + 36*n + 63)*x^3 - (16*n + 48)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(-n^3 + 5*n^2 + 13*n - 20 + (n^3 + 25*n - 20)*x + (4*n^2 + 3*n + 14)*x^2 + (4*n + 8)*x^3)]];
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] = (-(3*n^3 - 11*n^2 - 30*n + 44 - (n^3 + 2*n^2 + 60*n - 51)*x - (2*n^3 + 6*n^2 + 17*n + 17)*x^2 - (8*n^2 + 2*n + 14)*x^3 - (8*n - 8)*x^4)*v[i - 1] + (5*n^2 + 5*n - 20 + (4*n^2 + 9*n - 56)*x - (5*n^2 + 16*n + 67)*x^2 - (4*n^2 + 36*n + 63)*x^3 - (16*n + 48)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(-n^3 + 5*n^2 + 13*n - 20 + (n^3 + 25*n - 20)*x + (4*n^2 + 3*n + 14)*x^2 + (4*n + 8)*x^3)); 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.