A185410 → reverse → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-5\,x^{2} + 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} - 2\,x^{3} - 4\,n x^{2} + 4\,n x^{3}\right)\,P_{n-2}'(x) + \left(2 - 6\,x + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 2 | 2 | 4 | |||||
| 3 | 4 | 20 | 3 | 27 | ||||
| 4 | 8 | 120 | 108 | 4 | 240 | |||
| 5 | 16 | 592 | 1548 | 464 | 5 | 2625 | ||
| 6 | 32 | 2656 | 15504 | 14032 | 1790 | 6 | 34020 | |
| 7 | 64 | 11328 | 128880 | 256960 | 105600 | 6516 | 7 | 509355 |
Compute more
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Technical details
- Node id
n-36edb24e07155eca· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-2fbd00207b225162
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 1; 2: 2, 2; 3: 4, 20, 3; 4: 8, 120, 108, 4; 5: 16, 592, 1548, 464, 5; 6: 32, 2656, 15504, 14032, 1790, 6; 7: 64, 11328, 128880, 256960, 105600, 6516, 7; 8: 128, 46976, 960192, 3602240, 3304400, 711792, 22904, 8; 9: 256, 191744, 6677184, 42900736, 73569200, 36298464, 4472776, 78656, 9;
%F FormulaThe row polynomials satisfy P(n,x) = -(4*n^2-12*n+5)*x^2*P(n-2,x) - ((4*n-2)*x^2-(4*n-2)*x^3)*d/dx P(n-2,x) + (2+(4*n-6)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 5)*x^2*P[n - 2, x] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*D[P[n - 2, x], x] + (2 + (4*n - 6)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 12*n + 5)*x^2*v[i - 2] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*deriv(v[i - 2], x) + (2 + (4*n - 6)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], 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]
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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
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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
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
A triangular decomposition of the double factorial numbers A001147.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A decomposition of the double factorials A001147.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A decomposition of the double factorials A001147.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
A triangular decomposition of the double factorial numbers A001147.?
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.