A142462 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(-5 - 6\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 18 | 18 | 1 | 37 | |||||
| 3 | 288 | 432 | 146 | 1 | 867 | ||||
| 4 | 6624 | 13248 | 7794 | 1170 | 1 | 28837 | |||
| 5 | 198720 | 496800 | 416160 | 127440 | 9362 | 1 | 1248483 | ||
| 6 | 7352640 | 22057920 | 23978880 | 11194560 | 1995858 | 74898 | 1 | 66654757 | |
| 7 | 323516160 | 1132306560 | 1513451520 | 952862400 | 278214048 | 30611952 | 599186 | 1 | 4231561827 |
Compute more
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Technical details
- Node id
n-159b44d7207b424e· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-d4f1df9ae9c004ef
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: 18, 18, 1; 3: 288, 432, 146, 1; 4: 6624, 13248, 7794, 1170, 1; 5: 198720, 496800, 416160, 127440, 9362, 1; 6: 7352640, 22057920, 23978880, 11194560, 1995858, 74898, 1; 7: 323516160, 1132306560, 1513451520, 952862400, 278214048, 30611952, 599186, 1; 8: 16499324160, 65997296640, 104686888320, 83070126720, 34031932704, 6610500288, 464571954, 4793490, 1; 9: 956960801280, 4306323605760, 7906087918080, 7575130886400, 4011482543040, 1138967555040, 152864157600, 7011720720, 38347922, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (7*n-5+(7*n-6)*x)*P(n-1,x) - (7*x+7*x^2)*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[(7*n - 5 + (7*n - 6)*x)*P[n - 1, x] - (7*x + 7*x^2)*D[P[n - 1, x], 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] = (7*n - 5 + (7*n - 6)*x)*v[i - 1] - (7*x + 7*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 - 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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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