A099097 → reverse → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-27 + 3\,x + 27\,n - 6\,n x + 3\,n^{2} x\right)\,P_{n}(x) = \left(-81 - 63\,x - 4\,x^{2} + 81\,n + 18\,n x - 2\,n x^{2} + 9\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-81\,x - 45\,x^{2} - 4\,x^{3} - 9\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | |||
| 2 | 3 | 3 | |||
| 3 | 9 | 2 | 11 | ||
| 4 | 27 | 12 | 39 | ||
| 5 | 81 | 54 | 3 | 138 | |
| 6 | 243 | 216 | 27 | 486 | |
| 7 | 729 | 810 | 162 | 4 | 1705 |
| 8 | 2187 | 2916 | 810 | 48 | 5961 |
Compute more
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Technical details
- Node id
n-8e2140e39437a3ed· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-eb52c46fd35a0a6d
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 3; 3: 9, 2; 4: 27, 12; 5: 81, 54, 3; 6: 243, 216, 27; 7: 729, 810, 162, 4; 8: 2187, 2916, 810, 48; 9: 6561, 10206, 3645, 360, 5; 10: 19683, 34992, 15309, 2160, 75;
%F FormulaThe row polynomials satisfy (27*n-27+(3*n^2-6*n+3)*x)*P(n,x) = (81*n-81+(9*n^2+18*n-63)*x+(2*n^2-2*n-4)*x^2)*P(n-1,x) - (81*x+(9*n+45)*x^2+(4*n+4)*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[((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*D[P[n - 1, x], x])/(27*n - 27 + (3*n^2 - 6*n + 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] = ((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*v[i - 1] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*deriv(v[i - 1], x))/(27*n - 27 + (3*n^2 - 6*n + 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 multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
reversal of Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (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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from
Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (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
reversal of Riordan array (1, 3+x).?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.