basis-map coefficients of A026729, shifted → basis A013609 → D+1
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-3 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(1 - 4\,x^{2} - 2\,n - 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | -3 | -2 | -5 | ||||||
| 3 | 5 | 12 | 4 | 21 | |||||
| 4 | -7 | -30 | -36 | -8 | -81 | ||||
| 5 | 9 | 56 | 120 | 96 | 16 | 297 | |||
| 6 | -11 | -90 | -280 | -400 | -240 | -32 | -1053 | ||
| 7 | 13 | 132 | 540 | 1120 | 1200 | 576 | 64 | 3645 | |
| 8 | -15 | -182 | -924 | -2520 | -3920 | -3360 | -1344 | -128 | -12393 |
Compute more
?
Technical details
- Node id
n-5560650c49185322· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-4be2d3b20c0d29e0
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 -1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -3, -2; 3: 5, 12, 4; 4: -7, -30, -36, -8; 5: 9, 56, 120, 96, 16; 6: -11, -90, -280, -400, -240, -32; 7: 13, 132, 540, 1120, 1200, 576, 64; 8: -15, -182, -924, -2520, -3920, -3360, -1344, -128; 9: 17, 240, 1456, 4928, 10080, 12544, 8960, 3072, 256; 10: -19, -306, -2160, -8736, -22176, -36288, -37632, -23040, -6912, -512;
%F FormulaThe row polynomials satisfy (2*n-3+2*x)*P(n,x) = -(2*n-1+4*n*x+4*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + 4*n*x + 4*x^2)*P[n - 1, x])/(2*n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 1 + 4*n*x + 4*x^2)*v[i - 1])/(2*n - 3 + 2*x)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: 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 ≥ 1, 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
basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
Related ?
Same fingerprint ?
- A156992 → basis A013611 → D+1
- A013609 → basis A013610 → D+1
- A134082 → basis A013609
- A136572 → basis A013609 → D+1
- A085881 → basis A013611 → D+1
- A178821 → basis A013611 → D+1
- A038208 → basis A013611 → D+1
- A120909 → basis A013610 → D+1
- A178822 → basis A013611 → D+1
- A013609 → D → D+1
- A085880 → basis A013611 → D+1
- A130123 → basis A013609 → D+1
- basis-map coefficients of A026729 → basis A013609 → D+1
- A038221 → basis A013611 → D+1
- basis-map coefficients of A030528, shifted → basis A013609 → D+1
- A196347 → basis A013611 → D+1
- basis-map coefficients of A112227 → basis A013609 → D+1
- A120909 → D+1
- A356546 → basis A013611 → D+1
- A178820 → basis A013611 → D+1
- A028326 → basis A013611 → D+1
- A013609 → D+1
References
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.