A134346 → D+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-x + x^{2} - n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-2\,x - 6\,x^{2} - 6\,x^{3} - 2\,x^{4} - 2\,n - 8\,n x - 10\,n x^{2} - 4\,n x^{3} - 2\,n^{2} - 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-3 - 3\,x + 3\,x^{2} + 3\,x^{3} + 6\,n x + 6\,n x^{2} + 3\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 3 | 9 | ||||||
| 2 | 21 | 28 | 7 | 56 | |||||
| 3 | 60 | 135 | 90 | 15 | 300 | ||||
| 4 | 155 | 496 | 558 | 248 | 31 | 1488 | |||
| 5 | 378 | 1575 | 2520 | 1890 | 630 | 63 | 7056 | ||
| 6 | 889 | 4572 | 9525 | 10160 | 5715 | 1524 | 127 | 32512 | |
| 7 | 2040 | 12495 | 32130 | 44625 | 35700 | 16065 | 3570 | 255 | 146880 |
Compute more
?
Technical details
- Node id
n-b93164230fee0958· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-e99c3824c4864868
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 6, 3; 2: 21, 28, 7; 3: 60, 135, 90, 15; 4: 155, 496, 558, 248, 31; 5: 378, 1575, 2520, 1890, 630, 63; 6: 889, 4572, 9525, 10160, 5715, 1524, 127; 7: 2040, 12495, 32130, 44625, 35700, 16065, 3570, 255; 8: 4599, 32704, 100156, 171696, 178850, 114464, 42924, 8176, 511; 9: 10230, 82863, 294624, 601524, 773388, 644490, 343728, 110484, 18414, 1023;
%F FormulaThe row polynomials satisfy (n^2-n+(2*n-1)*x+x^2)*P(n,x) = -(2*n^2+2*n+(4*n^2+8*n+2)*x+(2*n^2+10*n+6)*x^2+(4*n+6)*x^3+2*x^4)*P(n-2,x) + (3*n^2-3+(3*n^2+6*n-3)*x+(6*n+3)*x^2+3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 2*n + (4*n^2 + 8*n + 2)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n + 6)*x^3 + 2*x^4)*P[n - 2, x] + (3*n^2 - 3 + (3*n^2 + 6*n - 3)*x + (6*n + 3)*x^2 + 3*x^3)*P[n - 1, x])/(n^2 - n + (2*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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 + 2*n + (4*n^2 + 8*n + 2)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n + 6)*x^3 + 2*x^4)*v[i - 2] + (3*n^2 - 3 + (3*n^2 + 6*n - 3)*x + (6*n + 3)*x^2 + 3*x^3)*v[i - 1])/(n^2 - n + (2*n - 1)*x + x^2)); 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 differential operators
-
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
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Geometric basis map (elementary) (above)
-
from
Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
Related ?
Same fingerprint ?
- A093375 → D
- A178821 → basis A013609 → θ+1
- basis-map coefficients of A127647 → D+1
- A038208 → D+1
- A196347 → θ+1
- A356546 → basis A013609 → D+1
- A028326 → basis A013609 → θ+1
- A028326 → basis A013609 → D+1
- A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
- A120910 → basis A013610 → D+1
- A038325 → basis A013609 → θ+1
- A003506 → D → θ+1
- A344565 → D+1
- A038338 → basis A013609 → θ+1
- A178822 → basis A013609 → D+1
- A124860 → D+1
- A178820 → basis A013609 → D+1
- A196347 → D+1
- A196347 → basis A013609 → θ+1
- A376827 → D+1
- A135838 → D+1
- A178821 → basis A013609 → D+1
- A085880 → θ+1
- A178820 → D+1
- A131431 → x ↦ x+1 → θ+1
- A093375 → D → D
- A134346 → θ+1
- A156992 → θ+1
- A038208 → D → θ+1
- basis-map coefficients of A127647 → θ+1
- A131431 → x ↦ x+1 → D+1
- basis-map coefficients of A113214, shifted → basis A013609 → D+1
- A356546 → θ+1
- A376827 → θ+1
- A038325 → θ+1
- A178821 → D+1
- A156992 → D+1
- A028326 → D → θ+1
- A120910 → basis A013610 → θ+1
- A178820 → basis A013609 → θ+1
- A196347 → basis A013609 → D+1
- A131431 → D+1 → basis A007318
- A038221 → basis A013609 → θ+1
- A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
- basis-map coefficients of A112227 → x ↦ x+1 → θ+1
- A028326 → θ+1
- A135065 → θ+1
- basis-map coefficients of A026729, shifted → basis A007318 → D+1
- A085880 → basis A013609 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
- A038338 → θ+1
- basis-map coefficients of A113214, shifted → θ+1
- A038330 → basis A013611 → θ+1
- A038338 → D+1
- A038208 → θ+1
- basis-map coefficients of A113214, shifted → basis A013609 → θ+1
- A178820 → θ+1
- A085881 → basis A013609 → θ+1
- A038221 → basis A013609 → D+1
- A038338 → basis A013609 → D+1
- A085880 → D → D+1
- A085881 → θ+1
- A178822 → θ+1
- A156992 → basis A013609 → D+1
- basis-map coefficients of A026729 → basis A007318 → D+1
- A085880 → D → θ+1
- A038330 → basis A013611 → D+1
- basis-map coefficients of A026729 → basis A007318 → θ+1
- A085881 → basis A013609 → D+1
- A038325 → basis A013609 → D+1
- A028326 → D+1
- A085881 → D+1
- A356546 → basis A013609 → θ+1
- A038208 → D → D+1
- A038221 → θ+1
- A356546 → D+1
- A038325 → D+1
- A135838 → θ+1
- basis-map coefficients of A112227 → x ↦ x+1 → D+1
- basis-map coefficients of A113214, shifted → D → θ+1
- A178821 → θ+1
- A135065 → D+1
- A027465 → basis A013611 → D+1
- A156992 → basis A013609 → θ+1
- A094305 → θ+1
- A038221 → D+1
- A085880 → D+1
- A027465 → basis A013611 → θ+1
- A178822 → basis A013609 → θ+1
- A178822 → D+1
- A094305 → D+1
- A028326 → D → D+1
- basis-map coefficients of A113214, shifted → D+1
- basis-map coefficients of A113214, shifted → D → D+1
- A085880 → basis A013609 → D+1
- A344565 → θ+1
- basis-map coefficients of A026729, shifted → basis A007318 → θ+1
- A216973 → θ+1
- A124860 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → D+1
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.