A133080 → θ+1 → reverse
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-1 - n x + n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(x + n x^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | Pn(1) |
|---|---|---|---|
| 0 | 1 | 1 | |
| 1 | 2 | 1 | 3 |
| 2 | 3 | 3 | |
| 3 | 4 | 3 | 7 |
| 4 | 5 | 5 | |
| 5 | 6 | 5 | 11 |
| 6 | 7 | 7 | |
| 7 | 8 | 7 | 15 |
| 8 | 9 | 9 | |
| 9 | 10 | 9 | 19 |
| 10 | 11 | 11 | |
| 11 | 12 | 11 | 23 |
| 12 | 13 | 13 | |
| 13 | 14 | 13 | 27 |
| 14 | 15 | 15 |
Compute more
?
Technical details
- Node id
n-3524478ec50f6156· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-6b183ed9136cbe2a
Observed on the stored rows ?
- Degrees: 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 (irregular)
- 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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 3; 3: 4, 3; 4: 5; 5: 6, 5; 6: 7; 7: 8, 7; 8: 9; 9: 10, 9;
%F FormulaThe row polynomials satisfy (n^2-n)*P(n,x) = (n^2-1+(n^2-n)*x)*P(n-1,x) - ((2*n^2-1)*x+(n^2-n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 1 + (n^2 - n)*x)*P[n - 1, x] - ((2*n^2 - 1)*x + (n^2 - n)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 1 + (n^2 - n)*x)*v[i - 1] - ((2*n^2 - 1)*x + (n^2 - n)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.