A201637 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(-3 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||
| 2 | 1 | 1 | ||||||
| 3 | 3 | 2 | 5 | |||||
| 4 | 15 | 20 | 6 | 41 | ||||
| 5 | 105 | 210 | 130 | 24 | 469 | |||
| 6 | 945 | 2520 | 2380 | 924 | 120 | 6889 | ||
| 7 | 10395 | 34650 | 44100 | 26432 | 7308 | 720 | 123605 | |
| 8 | 135135 | 540540 | 866250 | 705320 | 303660 | 64224 | 5040 | 2620169 |
Compute more
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Technical details
- Node id
n-984b9e7687210001· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-0c4bf0a352e1721b
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1; 3: 3, 2; 4: 15, 20, 6; 5: 105, 210, 130, 24; 6: 945, 2520, 2380, 924, 120; 7: 10395, 34650, 44100, 26432, 7308, 720; 8: 135135, 540540, 866250, 705320, 303660, 64224, 5040; 9: 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320; 10: 34459425, 183783600, 416215800, 520059540, 389449060, 177331440, 47324376, 6636960, 362880;
%F FormulaThe row polynomials satisfy P(n,x) = (2*n-3+(2*n-4)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + (2*n - 4)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + (2*n - 4)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
Related ?
Same fingerprint ?
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