A274498 → x ↦ x+1 → θ+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(18 + 9\,x + 6\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(54 - 45\,x - 36\,x^{2} - 4\,x^{3} + 90\,n x + 15\,n x^{2} - 2\,n x^{3} + 27\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-108\,x - 144\,x^{2} - 60\,x^{3} - 8\,x^{4} - 54\,n x^{2} - 54\,n x^{3} - 12\,n x^{4} - 6\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 3 | 3 | |||
| 2 | 9 | 12 | 21 | ||
| 3 | 27 | 36 | 63 | ||
| 4 | 81 | 216 | 108 | 405 | |
| 5 | 243 | 648 | 324 | 1215 | |
| 6 | 729 | 2916 | 2916 | 864 | 7425 |
| 7 | 2187 | 8748 | 8748 | 2592 | 22275 |
Compute more
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Technical details
- Node id
n-9f1d02e4eb03fbe6· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-ee9e93d4fe03de68
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 3; 2: 9, 12; 3: 27, 36; 4: 81, 216, 108; 5: 243, 648, 324; 6: 729, 2916, 2916, 864; 7: 2187, 8748, 8748, 2592; 8: 6561, 34992, 52488, 31104, 6480; 9: 19683, 104976, 157464, 93312, 19440;
%F FormulaThe row polynomials satisfy (18+(6*n+9)*x+(n^2+n)*x^2)*P(n,x) = (54+(90*n-45)*x+(27*n^2+15*n-36)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (108*x+(54*n+144)*x^2+(6*n^2+54*n+60)*x^3+(4*n^2+12*n+8)*x^4)*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[((54 + (90*n - 45)*x + (27*n^2 + 15*n - 36)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (108*x + (54*n + 144)*x^2 + (6*n^2 + 54*n + 60)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(18 + (6*n + 9)*x + (n^2 + n)*x^2)]];
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] = ((54 + (90*n - 45)*x + (27*n^2 + 15*n - 36)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (108*x + (54*n + 144)*x^2 + (6*n^2 + 54*n + 60)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(18 + (6*n + 9)*x + (n^2 + n)*x^2)); 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 ›
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.