A274498 → x ↦ x+1 → D+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-4 + 2\,x + 4\,x^{2} + 3\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-12 - 6\,x - 8\,x^{3} + 3\,n + 18\,n x + 4\,n x^{2} + 8\,n x^{3} + 9\,n^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-36 - 96\,x - 108\,x^{2} - 64\,x^{3} - 16\,x^{4} - 18\,n - 42\,n x - 44\,n x^{2} - 16\,n x^{3} - 6\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 3 | 3 | |||
| 2 | 15 | 6 | 21 | ||
| 3 | 45 | 18 | 63 | ||
| 4 | 189 | 180 | 36 | 405 | |
| 5 | 567 | 540 | 108 | 1215 | |
| 6 | 2187 | 3402 | 1620 | 216 | 7425 |
| 7 | 6561 | 10206 | 4860 | 648 | 22275 |
Compute more
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Technical details
- Node id
n-f9cf723e72635276· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-376115f4132a4e54
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: 15, 6; 3: 45, 18; 4: 189, 180, 36; 5: 567, 540, 108; 6: 2187, 3402, 1620, 216; 7: 6561, 10206, 4860, 648; 8: 24057, 52488, 40824, 12960, 1296; 9: 72171, 157464, 122472, 38880, 3888;
%F FormulaThe row polynomials satisfy (n^2+3*n-4+(4*n+2)*x+4*x^2)*P(n,x) = (9*n^2+3*n-12+(2*n^3+4*n^2+18*n-6)*x+(8*n^2+4*n)*x^2+(8*n-8)*x^3)*P(n-1,x) - (18*n+36+(6*n^2+42*n+96)*x+(4*n^2+44*n+108)*x^2+(16*n+64)*x^3+16*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[((9*n^2 + 3*n - 12 + (2*n^3 + 4*n^2 + 18*n - 6)*x + (8*n^2 + 4*n)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (18*n + 36 + (6*n^2 + 42*n + 96)*x + (4*n^2 + 44*n + 108)*x^2 + (16*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n - 4 + (4*n + 2)*x + 4*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] = ((9*n^2 + 3*n - 12 + (2*n^3 + 4*n^2 + 18*n - 6)*x + (8*n^2 + 4*n)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (18*n + 36 + (6*n^2 + 42*n + 96)*x + (4*n^2 + 44*n + 108)*x^2 + (16*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n - 4 + (4*n + 2)*x + 4*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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] 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.