A153653 → basis A007318
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(-646 - 646\,x + 323\,n + 323\,n x\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 38 + 19\,x\)
\(P_{2}(x) = 722 + 722\,x + 2\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 2 | 2 | |||||||
| 1 | 38 | 19 | 57 | ||||||
| 2 | 722 | 722 | 2 | 1446 | |||||
| 3 | 13718 | 20577 | 6863 | 2 | 41160 | ||||
| 4 | 260642 | 521284 | 268155 | 7513 | 2 | 1057596 | |||
| 5 | 4952198 | 12380495 | 9920714 | 2500576 | 8163 | 2 | 29762148 | ||
| 6 | 94091762 | 282275286 | 287210720 | 103962630 | 4944247 | 8813 | 2 | 772493460 | |
| 7 | 1787743478 | 6257102173 | 8059987233 | 4507212650 | 924173821 | 7599168 | 9463 | 2 | 21543827988 |
Compute more
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Technical details
- Node id
n-3a242ada054de5d6· revision 0 · JSON- Definition source
derived:basis_map_A007318- Fingerprint
f6-9b155509a8b144c4
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: 2; 1: 38, 19; 2: 722, 722, 2; 3: 13718, 20577, 6863, 2; 4: 260642, 521284, 268155, 7513, 2; 5: 4952198, 12380495, 9920714, 2500576, 8163, 2; 6: 94091762, 282275286, 287210720, 103962630, 4944247, 8813, 2; 7: 1787743478, 6257102173, 8059987233, 4507212650, 924173821, 7599168, 9463, 2; 8: 33967126082, 135868504328, 206321056577, 143423404583, 41532481563, 2539210537, 10465339, 10113, 2; 9: 645375395558, 2904189280011, 5172930495620, 4552373427952, 1980826197842, 347573578084, 5017729028, 13542760, 10763, 2;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (323*n-646+(323*n-646)*x)*P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 38 + 19*x, P(2,x) = 722 + 722*x + 2*x^2.
%t MathematicaP[0, x_] := 2;
P[1, x_] := 38 + 19*x;
P[2, x_] := 722 + 722*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((323*n - 646 + (323*n - 646)*x)*P[n - 2, x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 38 + 19*x; v[3] = 722 + 722*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((323*n - 646 + (323*n - 646)*x)*v[i - 2] + (2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); 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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and…?
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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 T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and…?
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