basis-map coefficients of A026729, shifted → basis A013610 → D+1
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-5 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(2 + 3\,x - 9\,x^{2} - 3\,n - 9\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | -4 | -3 | -7 | |||||||||||||
| 3 | 7 | 24 | 9 | 40 | ||||||||||||
| 4 | -10 | -63 | -108 | -27 | -208 | |||||||||||
| 5 | 13 | 120 | 378 | 432 | 81 | 1024 | ||||||||||
| 6 | -16 | -195 | -900 | -1890 | -1620 | -243 | -4864 | |||||||||
| 7 | 19 | 288 | 1755 | 5400 | 8505 | 5832 | 729 | 22528 | ||||||||
| 8 | -22 | -399 | -3024 | -12285 | -28350 | -35721 | -20412 | -2187 | -102400 | |||||||
| 9 | 25 | 528 | 4788 | 24192 | 73710 | 136080 | 142884 | 69984 | 6561 | 458752 | ||||||
| 10 | -28 | -675 | -7128 | -43092 | -163296 | -398034 | -612360 | -551124 | -236196 | -19683 | -2031616 | |||||
| 11 | 31 | 840 | 10125 | 71280 | 323190 | 979776 | 1990170 | 2624400 | 2066715 | 787320 | 59049 | 8912896 | ||||
| 12 | -34 | -1023 | -13860 | -111375 | -588060 | -2133054 | -5388768 | -9382230 | -10825650 | -7577955 | -2598156 | -177147 | -38797312 | |||
| 13 | 37 | 1224 | 18414 | 166320 | 1002375 | 4234032 | 12798324 | 27713664 | 42220035 | 43302600 | 27280638 | 8503056 | 531441 | 167772160 | ||
| 14 | -40 | -1443 | -23868 | -239382 | -1621620 | -7818525 | -27521208 | -71304948 | -135104112 | -182953485 | -168880140 | -96722262 | -27634932 | -1594323 | -721420288 | |
| 15 | 43 | 1680 | 30303 | 334152 | 2513511 | 13621608 | 54729675 | 165127248 | 374350977 | 630485856 | 768404637 | 644815080 | 338527917 | 89282088 | 4782969 | 3087007744 |
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Technical details
- Node id
n-1c2a0f0db3be1b21· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-1f45c55ca90e4ac8
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: -4, -3; 3: 7, 24, 9; 4: -10, -63, -108, -27; 5: 13, 120, 378, 432, 81; 6: -16, -195, -900, -1890, -1620, -243; 7: 19, 288, 1755, 5400, 8505, 5832, 729; 8: -22, -399, -3024, -12285, -28350, -35721, -20412, -2187; 9: 25, 528, 4788, 24192, 73710, 136080, 142884, 69984, 6561; 10: -28, -675, -7128, -43092, -163296, -398034, -612360, -551124, -236196, -19683;
%F FormulaThe row polynomials satisfy (3*n-5+3*x)*P(n,x) = -(3*n-2+(9*n-3)*x+9*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[(-(3*n - 2 + (9*n - 3)*x + 9*x^2)*P[n - 1, x])/(3*n - 5 + 3*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] = (-(3*n - 2 + (9*n - 3)*x + 9*x^2)*v[i - 1])/(3*n - 5 + 3*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_A013610 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 …?
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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 …?
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from
basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
Related ?
Same fingerprint ?
- A013610 → D → D+1
- A013610 → basis A013611 → D+1
- A136572 → basis A013610 → D+1
- A013610 → D+1
- A130123 → basis A013610 → D+1
- basis-map coefficients of A026729 → basis A013610 → D+1
- basis-map coefficients of A112227 → basis A013610 → D+1
- basis-map coefficients of A030528, shifted → basis A013610 → 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.