A143685 → γ → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n + 9\,n^{2} x\right)\,P_{n}(x) = \left(-36\,x - 324\,x^{2} + n + 36\,n x + 162\,n x^{2} + 9\,n^{2} x + 162\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x - 54\,x^{2} - 648\,x^{3} - 9\,n x^{2} - 324\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 1 | 18 | 19 | ||
| 3 | 1 | 36 | 37 | ||
| 4 | 1 | 54 | 243 | 298 | |
| 5 | 1 | 72 | 729 | 802 | |
| 6 | 1 | 90 | 1458 | 2916 | 4465 |
| 7 | 1 | 108 | 2430 | 11664 | 14203 |
Compute more
?
Technical details
- Node id
n-b4e5e37f3793a6df· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-e0fc085c281e048d
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: 1, 18; 3: 1, 36; 4: 1, 54, 243; 5: 1, 72, 729; 6: 1, 90, 1458, 2916; 7: 1, 108, 2430, 11664; 8: 1, 126, 3645, 29160, 32805; 9: 1, 144, 5103, 58320, 164025;
%F FormulaThe row polynomials satisfy (n+9*n^2*x)*P(n,x) = (n+(9*n^2+36*n-36)*x+(162*n^2+162*n-324)*x^2)*P(n-1,x) - (x+(9*n+54)*x^2+(324*n+648)*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[((n + (9*n^2 + 36*n - 36)*x + (162*n^2 + 162*n - 324)*x^2)*P[n - 1, x] - (x + (9*n + 54)*x^2 + (324*n + 648)*x^3)*D[P[n - 1, x], x])/(n + 9*n^2*x)]];
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] = ((n + (9*n^2 + 36*n - 36)*x + (162*n^2 + 162*n - 324)*x^2)*v[i - 1] - (x + (9*n + 54)*x^2 + (324*n + 648)*x^3)*deriv(v[i - 1], x))/(n + 9*n^2*x)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: 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 ≥ 0, 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
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
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.