A201730 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(96 + 88\,x - 48\,n - 60\,n x + 4\,n x^{2} + 8\,n^{2} x - 6\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(240 + 280\,x + 88\,x^{2} - 120\,n - 168\,n x - 50\,n x^{2} - 2\,n x^{3} + 14\,n^{2} x - 7\,n^{2} x^{2} - n^{2} x^{3} + 5\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-72 - 180\,x - 64\,x^{2} + 44\,x^{3} + 24\,n x + 22\,n x^{2} - 4\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x^{2} + 4\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | |||
| 2 | 2 | 2 | |||
| 3 | 5 | 2 | 7 | ||
| 4 | 14 | 12 | 26 | ||
| 5 | 41 | 52 | 3 | 96 | |
| 6 | 122 | 200 | 30 | 352 | |
| 7 | 365 | 726 | 189 | 4 | 1284 |
| 8 | 1094 | 2548 | 966 | 56 | 4664 |
Compute more
?
Technical details
- Node id
n-ea8bb83df35fc654· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-d2fcce66244b0ee8
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 -1)
- 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: 1; 2: 2; 3: 5, 2; 4: 14, 12; 5: 41, 52, 3; 6: 122, 200, 30; 7: 365, 726, 189, 4; 8: 1094, 2548, 966, 56; 9: 3281, 8744, 4386, 464, 5; 10: 9842, 29520, 18468, 2976, 90;
%F FormulaThe row polynomials satisfy -(48*n-96-(8*n^2-60*n+88)*x-(2*n^3-6*n^2+4*n)*x^2)*P(n,x) = -(120*n-240-(14*n^2-168*n+280)*x-(5*n^3-7*n^2-50*n+88)*x^2-(n^3-n^2-2*n)*x^3)*P(n-1,x) - (72-(24*n-180)*x-(6*n^2+22*n-64)*x^2-(4*n^2-4*n+44)*x^3+(2*n^2+2*n)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 240 - (14*n^2 - 168*n + 280)*x - (5*n^3 - 7*n^2 - 50*n + 88)*x^2 - (n^3 - n^2 - 2*n)*x^3)*P[n - 1, x] - (72 - (24*n - 180)*x - (6*n^2 + 22*n - 64)*x^2 - (4*n^2 - 4*n + 44)*x^3 + (2*n^2 + 2*n)*x^4)*D[P[n - 1, x], x])/(-48*n + 96 + (8*n^2 - 60*n + 88)*x + (2*n^3 - 6*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(120*n - 240 - (14*n^2 - 168*n + 280)*x - (5*n^3 - 7*n^2 - 50*n + 88)*x^2 - (n^3 - n^2 - 2*n)*x^3)*v[i - 1] - (72 - (24*n - 180)*x - (6*n^2 + 22*n - 64)*x^2 - (4*n^2 - 4*n + 44)*x^3 + (2*n^2 + 2*n)*x^4)*deriv(v[i - 1], x))/(-48*n + 96 + (8*n^2 - 60*n + 88)*x + (2*n^3 - 6*n^2 + 4*n)*x^2)); 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 multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.