A144445 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(7 + 7\,x\right)\,P_{n-2}(x) + \left(-5 - 6\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 25 | 25 | 1 | 51 | |||||
| 3 | 414 | 621 | 209 | 1 | 1245 | ||||
| 4 | 9697 | 19394 | 11378 | 1681 | 1 | 42151 | |||
| 5 | 293808 | 734520 | 614526 | 187269 | 13457 | 1 | 1843581 | ||
| 6 | 10938775 | 32816325 | 35650582 | 16607289 | 2941922 | 107665 | 1 | 99062559 | |
| 7 | 483362756 | 1691769646 | 2260348482 | 1421447090 | 413836192 | 45192021 | 861329 | 1 | 6316817517 |
Compute more
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Technical details
- Node id
n-07cdb61aa05a9163· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-9024966cc99eeda3
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: 25, 25, 1; 3: 414, 621, 209, 1; 4: 9697, 19394, 11378, 1681, 1; 5: 293808, 734520, 614526, 187269, 13457, 1; 6: 10938775, 32816325, 35650582, 16607289, 2941922, 107665, 1; 7: 483362756, 1691769646, 2260348482, 1421447090, 413836192, 45192021, 861329, 1; 8: 24728071981, 98912287924, 156858256961, 124381763149, 50877042503, 9848815669, 686385938, 6890641, 1; 9: 1437611714190, 6469252713855, 11875042529380, 11372802854840, 6016901731428, 1705170190465, 227978493176, 10363834149, 55125137, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (7+7*x)*P(n-2,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 >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(7 + 7*x)*P[n - 2, x] + (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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (7 + 7*x)*v[i - 2] + (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 ›
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from
Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
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