A166960 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + x^{3} - 2\,n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - n x^{3}\right)\,P_{n-2}''(x) + \left(1 + 2\,n x\right)\,P_{n-1}(x) + \left(3\,x - x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 12 | 3 | 16 | |||||
| 3 | 1 | 54 | 63 | 4 | 122 | ||||
| 4 | 1 | 224 | 810 | 232 | 5 | 1272 | |||
| 5 | 1 | 906 | 8634 | 6952 | 705 | 6 | 17204 | ||
| 6 | 1 | 3636 | 84501 | 157280 | 43695 | 1908 | 7 | 291028 | |
| 7 | 1 | 14558 | 793233 | 3079820 | 1878775 | 225990 | 4795 | 8 | 5997180 |
Compute more
?
Technical details
- Node id
n-0aa890aa63c05998· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-813e3a68dec52ec0
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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: 1, 2; 2: 1, 12, 3; 3: 1, 54, 63, 4; 4: 1, 224, 810, 232, 5; 5: 1, 906, 8634, 6952, 705, 6; 6: 1, 3636, 84501, 157280, 43695, 1908, 7; 7: 1, 14558, 793233, 3079820, 1878775, 225990, 4795, 8; 8: 1, 58248, 7291956, 55622984, 66218250, 17153760, 1025920, 11472, 9; 9: 1, 233010, 66326580, 958026000, 2072628630, 1014844860, 130320960, 4251648, 26541, 10;
%F FormulaThe row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-2,x) - ((2*n+2)*x^2-(n+1)*x^3)*d/dx P(n-2,x) - (n+1)*x^3*d^2/dx^2 P(n-2,x) + (1+2*n*x)*P(n-1,x) + (3*x-x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 2, x] - ((2*n + 2)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] - (n + 1)*x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*n*x)*P[n - 1, x] + (3*x - x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 2] - ((2*n + 2)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) - (n + 1)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*n*x)*v[i - 1] + (3*x - x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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 multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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.