A099093 → x ↦ x+1 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n\right)\,P_{n}(x) = \left(-5\,x - 7\,x^{2} + 3\,n + 8\,n x + 7\,n x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 10\,x^{2} - 7\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 3 | 6 | ||||||
| 2 | 9 | 21 | 12 | 42 | |||||
| 3 | 27 | 99 | 117 | 45 | 288 | ||||
| 4 | 81 | 405 | 738 | 585 | 171 | 1980 | |||
| 5 | 243 | 1539 | 3807 | 4617 | 2754 | 648 | 13608 | ||
| 6 | 729 | 5589 | 17496 | 28701 | 26082 | 12474 | 2457 | 93528 | |
| 7 | 2187 | 19683 | 74601 | 154629 | 189621 | 137781 | 54999 | 9315 | 642816 |
Compute more
?
Technical details
- Node id
n-574e6ccfb35bcaa6· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-c912b16b18733d86
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: 3, 3; 2: 9, 21, 12; 3: 27, 99, 117, 45; 4: 81, 405, 738, 585, 171; 5: 243, 1539, 3807, 4617, 2754, 648; 6: 729, 5589, 17496, 28701, 26082, 12474, 2457; 7: 2187, 19683, 74601, 154629, 189621, 137781, 54999, 9315; 8: 6561, 67797, 301806, 756945, 1171341, 1146555, 694008, 237735, 35316; 9: 19683, 229635, 1174419, 3459105, 6472548, 7986438, 6503895, 3373569, 1012095, 133893;
%F FormulaThe row polynomials satisfy n*P(n,x) = (3*n+(8*n-5)*x+(7*n-7)*x^2)*P(n-1,x) - (3*x+10*x^2+7*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + (8*n - 5)*x + (7*n - 7)*x^2)*P[n - 1, x] - (3*x + 10*x^2 + 7*x^3)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + (8*n - 5)*x + (7*n - 7)*x^2)*v[i - 1] - (3*x + 10*x^2 + 7*x^3)*deriv(v[i - 1], x))/(n)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A007318 of Riordan array (1, 3+3x).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Riordan array (1, 3+3x).?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
-
from
Riordan array (1, 3+3x).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Riordan array (1, 3+3x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Riordan array (1, 3+3x).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Riordan array (1, 3+3x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Riordan array (1, 3+3x).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1, 3+3x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Riordan array (1, 3+3x).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1, 3+3x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Riordan array (1, 3+3x).?
-
-
from
basis_map_A007318 of Riordan array (1, 3+3x).?
Related ?
Same fingerprint ?
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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741