A209414 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-3 + 4\,n\right)\,P_{n}(x) = \left(-21 - 23\,x + 28\,n + 32\,n x - 5\,n^{2} - 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-18 - 55\,x - 36\,x^{2} + 9\,n + 26\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x) + \left(-9\,x - 17\,x^{2} - 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 6 | 6 | 1 | 13 | |||||
| 3 | 18 | 28 | 12 | 1 | 59 | ||||
| 4 | 54 | 114 | 80 | 20 | 1 | 269 | |||
| 5 | 162 | 432 | 422 | 180 | 30 | 1 | 1227 | ||
| 6 | 486 | 1566 | 1962 | 1190 | 350 | 42 | 1 | 5597 | |
| 7 | 1458 | 5508 | 8424 | 6640 | 2828 | 616 | 56 | 1 | 25531 |
Compute more
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Technical details
- Node id
n-e3b686c2ee3ca8b9· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-d147b3e76f252408
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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: 2, 1; 2: 6, 6, 1; 3: 18, 28, 12, 1; 4: 54, 114, 80, 20, 1; 5: 162, 432, 422, 180, 30, 1; 6: 486, 1566, 1962, 1190, 350, 42, 1; 7: 1458, 5508, 8424, 6640, 2828, 616, 56, 1; 8: 4374, 18954, 34236, 33240, 18612, 5964, 1008, 72, 1; 9: 13122, 64152, 133650, 154260, 107252, 45648, 11508, 1560, 90, 1;
%F FormulaThe row polynomials satisfy (4*n-3)*P(n,x) = -(5*n^2-28*n+21+(8*n^2-32*n+23)*x)*P(n-1,x) + (9*n-18+(26*n-55)*x+(16*n-36)*x^2)*d/dx P(n-1,x) - (9*x+17*x^2+8*x^3)*d^2/dx^2 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[(-(5*n^2 - 28*n + 21 + (8*n^2 - 32*n + 23)*x)*P[n - 1, x] + (9*n - 18 + (26*n - 55)*x + (16*n - 36)*x^2)*D[P[n - 1, x], x] - (9*x + 17*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(4*n - 3)]];
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] = (-(5*n^2 - 28*n + 21 + (8*n^2 - 32*n + 23)*x)*v[i - 1] + (9*n - 18 + (26*n - 55)*x + (16*n - 36)*x^2)*deriv(v[i - 1], x) - (9*x + 17*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n - 3)); 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 (−∞, -1]
- 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)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
Related ?
Candidate equalities ?
- A209414 → basis A007318 15 rows; no all-index equality certificate was found
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