A266213 → basis A013610 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-18 + 9\,n\right)\,P_{n}(x) = \left(18 - 45\,n + 18\,n^{2}\right)\,P_{n-1}(x) + \left(375 + 423\,x + 108\,x^{2} - 123\,n - 126\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(164 + 291\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 9 + 6\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 9 | 6 | 15 | |||||
| 3 | 49 | 60 | 18 | 127 | ||||
| 4 | 281 | 498 | 288 | 54 | 1121 | |||
| 5 | 1601 | 3720 | 3186 | 1188 | 162 | 9857 | ||
| 6 | 9129 | 26238 | 29736 | 16578 | 4536 | 486 | 86703 | |
| 7 | 52049 | 178260 | 251298 | 186408 | 76626 | 16524 | 1458 | 762623 |
Compute more
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Technical details
- Node id
n-594bbffcdc3fa5ea· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-5d09cadc540fdad8
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 1; 2: 9, 6; 3: 49, 60, 18; 4: 281, 498, 288, 54; 5: 1601, 3720, 3186, 1188, 162; 6: 9129, 26238, 29736, 16578, 4536, 486; 7: 52049, 178260, 251298, 186408, 76626, 16524, 1458; 8: 296761, 1179786, 1988928, 1841454, 1010232, 328050, 58320, 4374; 9: 1692001, 7658400, 15023970, 16673580, 11441250, 4966920, 1331154, 201204, 13122;
%F FormulaThe row polynomials satisfy (9*n-18)*P(n,x) = (18*n^2-45*n+18)*P(n-1,x) - (123*n-375+(126*n-423)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (164+291*x+162*x^2+27*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 9 + 6*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 45*n + 18)*P[n - 1, x] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (164 + 291*x + 162*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n - 18)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 9 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = ((18*n^2 - 45*n + 18)*v[i - 1] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (164 + 291*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n - 18)); 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_A013610 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
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Geometric basis map (elementary) polytool ? step ›
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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from
basis_map_A013610 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
Related ?
Candidate equalities ?
- A266213 → basis A013610 → 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