A160562 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-8\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 4\,x^{3}\right)\,P_{n-2}''(x) + \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 20 | 3 | 24 | |||||
| 3 | 1 | 182 | 105 | 4 | 292 | ||||
| 4 | 1 | 1640 | 2898 | 336 | 5 | 4880 | |||
| 5 | 1 | 14762 | 74910 | 20328 | 825 | 6 | 110832 | ||
| 6 | 1 | 132860 | 1894893 | 1095952 | 92235 | 1716 | 7 | 3217664 | |
| 7 | 1 | 1195742 | 47571615 | 56228172 | 8840975 | 318318 | 3185 | 8 | 114158016 |
Compute more
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Technical details
- Node id
n-9718b3717ad8e9e3· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-98f5f68794f5eec6
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, 20, 3; 3: 1, 182, 105, 4; 4: 1, 1640, 2898, 336, 5; 5: 1, 14762, 74910, 20328, 825, 6; 6: 1, 132860, 1894893, 1095952, 92235, 1716, 7; 7: 1, 1195742, 47571615, 56228172, 8840975, 318318, 3185, 8; 8: 1, 10761680, 1191083988, 2818609248, 786404190, 49125648, 909636, 5440, 9; 9: 1, 96855122, 29793242220, 139699965136, 67222000950, 6887888436, 211041740, 2263584, 8721, 10;
%F FormulaThe row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) - (8*x+4*x^2)*d/dx P(n-2,x) - (4*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (2+2*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[-(1 + 2*x + x^2)*P[n - 2, x] - (8*x + 4*x^2)*D[P[n - 2, x], x] - (4*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*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] = -(1 + 2*x + x^2)*v[i - 2] - (8*x + 4*x^2)*deriv(v[i - 2], x) - (4*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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.