basis-map coefficients of A371080, shifted → reverse
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 3\,x + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 1 | 3 | 4 | |||||
| 3 | 1 | 9 | 18 | 28 | ||||
| 4 | 1 | 18 | 99 | 162 | 280 | |||
| 5 | 1 | 30 | 315 | 1350 | 1944 | 3640 | ||
| 6 | 1 | 45 | 765 | 6075 | 22194 | 29160 | 58240 | |
| 7 | 1 | 63 | 1575 | 19845 | 131544 | 428652 | 524880 | 1106560 |
Compute more
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Technical details
- Node id
n-6e131d90e56c0bf1· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-bc84b3931f09589a
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 1; 2: 1, 3; 3: 1, 9, 18; 4: 1, 18, 99, 162; 5: 1, 30, 315, 1350, 1944; 6: 1, 45, 765, 6075, 22194, 29160; 7: 1, 63, 1575, 19845, 131544, 428652, 524880; 8: 1, 84, 2898, 52920, 548289, 3191076, 9526572, 11022480; 9: 1, 108, 4914, 122472, 1818369, 16350012, 86112396, 239660208, 264539520;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(3*n-3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 3)*x)*P[n - 1, 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] = (1 + (3*n - 3)*x)*v[i - 1]); 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]
- 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)