A143685 → γ → basis A007318
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(n\right)\,P_{n}(x) = \left(-18 - 18\,x + 19\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-37 - 73\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 10 | 9 | 19 | ||
| 3 | 19 | 18 | 37 | ||
| 4 | 109 | 189 | 81 | 379 | |
| 5 | 280 | 522 | 243 | 1045 | |
| 6 | 1261 | 3204 | 2673 | 729 | 7867 |
| 7 | 3781 | 10422 | 9558 | 2916 | 26677 |
Compute more
?
Other recurrences
Image under the basis map A007318 · proved equal for all n (proof step)
\(P_{n}(x) = \left(9 + 9\,x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
- Node id
n-9957432edab6d129· revision 0 · JSON- Definition source
derived:shift_x_plus_onederived:basis_map_A007318(equal bye-c81b83be35fe8f46)- Fingerprint
f6-4f014acb052a88e2
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 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: 1; 2: 10, 9; 3: 19, 18; 4: 109, 189, 81; 5: 280, 522, 243; 6: 1261, 3204, 2673, 729; 7: 3781, 10422, 9558, 2916; 8: 15130, 50607, 62451, 33534, 6561; 9: 49159, 178434, 242271, 145800, 32805;
%F FormulaThe row polynomials satisfy n*P(n,x) = (19*n-18+(18*n-18)*x)*P(n-1,x) - (37+73*x+36*x^2)*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[((19*n - 18 + (18*n - 18)*x)*P[n - 1, x] - (37 + 73*x + 36*x^2)*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] = ((19*n - 18 + (18*n - 18)*x)*v[i - 1] - (37 + 73*x + 36*x^2)*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 (−∞, -1]
- 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)
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
gamma_polynomial of Pascal-(1,9,1) array.?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Pascal-(1,9,1) array.?
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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-(1,9,1) array.?
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from
gamma_polynomial of Pascal-(1,9,1) array.?
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