A249250 → x ↦ x+1 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - 2\,n + 3\,n^{2}\right)\,P_{n-1}(x) + \left(42 + 27\,x + 3\,x^{2} - 25\,n - 14\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(50 + 53\,x + 16\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 7 | 3 | 10 | ||||||
| 2 | 25 | 18 | 3 | 46 | |||||
| 3 | 103 | 105 | 33 | 3 | 244 | ||||
| 4 | 409 | 540 | 252 | 48 | 3 | 1252 | |||
| 5 | 1639 | 2655 | 1638 | 474 | 63 | 3 | 6472 | ||
| 6 | 6553 | 12582 | 9657 | 3756 | 771 | 78 | 3 | 33400 | |
| 7 | 26215 | 58197 | 53415 | 26097 | 7269 | 1143 | 93 | 3 | 172432 |
Compute more
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Technical details
- Node id
n-96ebc83d17e3d946· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-450e847a2555bc30
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: 7, 3; 2: 25, 18, 3; 3: 103, 105, 33, 3; 4: 409, 540, 252, 48, 3; 5: 1639, 2655, 1638, 474, 63, 3; 6: 6553, 12582, 9657, 3756, 771, 78, 3; 7: 26215, 58197, 53415, 26097, 7269, 1143, 93, 3; 8: 104857, 264240, 282234, 166044, 58500, 12552, 1590, 108, 3; 9: 419431, 1182795, 1440996, 991584, 422814, 115266, 19980, 2112, 123, 3;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (3*n^2-2*n-1)*P(n-1,x) - (25*n-42+(14*n-27)*x+(n-3)*x^2)*d/dx P(n-1,x) + (50+53*x+16*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 7 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n - 1)*P[n - 1, x] - (25*n - 42 + (14*n - 27)*x + (n - 3)*x^2)*D[P[n - 1, x], x] + (50 + 53*x + 16*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - 2*n - 1)*v[i - 1] - (25*n - 42 + (14*n - 27)*x + (n - 3)*x^2)*deriv(v[i - 1], x) + (50 + 53*x + 16*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 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 (−∞, -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)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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from
basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
Related ?
Candidate equalities ?
- A249250 → x ↦ x+1 → 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