A099097 → basis A013609 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(-36 - 62\,x - 26\,x^{2} + 36\,n + 40\,n x + 13\,n x^{2}\right)\,P_{n-1}(x) + \left(-36\,x - 44\,x^{2} - 13\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 6 | 3 | 9 | ||||||
| 3 | 36 | 38 | 10 | 84 | |||||
| 4 | 216 | 348 | 186 | 33 | 783 | ||||
| 5 | 1296 | 2808 | 2272 | 814 | 109 | 7299 | |||
| 6 | 7776 | 21168 | 22968 | 12420 | 3348 | 360 | 68040 | ||
| 7 | 46656 | 152928 | 208224 | 150776 | 61248 | 13236 | 1189 | 634257 | |
| 8 | 279936 | 1073088 | 1758240 | 1596432 | 867624 | 282276 | 50910 | 3927 | 5912433 |
Compute more
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Technical details
- Node id
n-c45ee837fbbe8408· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-f2c232606f089033
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): 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 6, 3; 3: 36, 38, 10; 4: 216, 348, 186, 33; 5: 1296, 2808, 2272, 814, 109; 6: 7776, 21168, 22968, 12420, 3348, 360; 7: 46656, 152928, 208224, 150776, 61248, 13236, 1189; 8: 279936, 1073088, 1758240, 1596432, 867624, 282276, 50910, 3927; 9: 1679616, 7371648, 14121216, 15422688, 10504816, 4569800, 1240008, 191906, 12970; 10: 10077696, 49828608, 109268352, 139489344, 114248064, 62264928, 22581624, 5255556, 712302, 42837;
%F FormulaThe row polynomials satisfy (6*n-6)*P(n,x) = (36*n-36+(40*n-62)*x+(13*n-26)*x^2)*P(n-1,x) - (36*x+44*x^2+13*x^3)*d/dx 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[((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*P[n - 1, x] - (36*x + 44*x^2 + 13*x^3)*D[P[n - 1, x], x])/(6*n - 6)]];
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] = ((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*v[i - 1] - (36*x + 44*x^2 + 13*x^3)*deriv(v[i - 1], x))/(6*n - 6)); 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]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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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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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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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ 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)
- ↑ 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)
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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ 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)
- ↑ 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)
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from
Riordan array (1, 3+x).?
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from
basis_map_A013609 of Riordan array (1, 3+x).?
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