A113953 → basis A013609 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n\right)\,P_{n}(x) = \left(-10\,x - 9\,x^{2} + 4\,n + 12\,n x + 9\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 20\,x^{2} - 9\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 4 | 8 | 3 | 15 | |||||
| 3 | 8 | 28 | 22 | 5 | 63 | ||||
| 4 | 16 | 80 | 112 | 60 | 11 | 279 | |||
| 5 | 32 | 208 | 432 | 376 | 146 | 21 | 1215 | ||
| 6 | 64 | 512 | 1424 | 1792 | 1132 | 352 | 43 | 5319 | |
| 7 | 128 | 1216 | 4256 | 7152 | 6424 | 3172 | 814 | 85 | 23247 |
Compute more
?
Technical details
- Node id
n-54aeb849ef21af75· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-a0b8d151a40b37b3
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: 2, 1; 2: 4, 8, 3; 3: 8, 28, 22, 5; 4: 16, 80, 112, 60, 11; 5: 32, 208, 432, 376, 146, 21; 6: 64, 512, 1424, 1792, 1132, 352, 43; 7: 128, 1216, 4256, 7152, 6424, 3172, 814, 85; 8: 256, 2816, 11904, 25280, 30016, 20880, 8472, 1860, 171; 9: 512, 6400, 31744, 81920, 122432, 111776, 63360, 21792, 4170, 341;
%F FormulaThe row polynomials satisfy 2*n*P(n,x) = (4*n+(12*n-10)*x+(9*n-9)*x^2)*P(n-1,x) - (4*x+20*x^2+9*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[((4*n + (12*n - 10)*x + (9*n - 9)*x^2)*P[n - 1, x] - (4*x + 20*x^2 + 9*x^3)*D[P[n - 1, x], x])/(2*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] = ((4*n + (12*n - 10)*x + (9*n - 9)*x^2)*v[i - 1] - (4*x + 20*x^2 + 9*x^3)*deriv(v[i - 1], x))/(2*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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A013609 of A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of A Jacobsthal triangle.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
A Jacobsthal triangle.?
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from
basis-map coefficients of A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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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
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Pascal-(1,2,1) array read by antidiagonals.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Pascal-(1,2,1) array read by antidiagonals.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
- ↑ 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)
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
- ↑ Ma–Wang interval theorem (above)
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from
A Jacobsthal triangle.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ 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)
- ↑ 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)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
- ↑ 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)
- ↑ 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)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
-
from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
- ↑ 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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ 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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
A Jacobsthal triangle.?
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from
basis_map_A013609 of A Jacobsthal triangle.?
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