A266213 → basis A013610 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(168 + 1026\,x + 1566\,x^{2} - 162\,x^{3} - 486\,x^{4} - 72\,n - 576\,n x - 2160\,n x^{2} - 486\,n x^{3} + 243\,n x^{4} - 270\,n^{2} x + 648\,n^{2} x^{2} + 243\,n^{2} x^{3} + 162\,n^{3} x\right)\,P_{n}(x) = \left(-168 - 1026\,x - 1566\,x^{2} + 162\,x^{3} + 486\,x^{4} - 96\,n + 774\,n x + 2916\,n x^{2} + 162\,n x^{3} - 729\,n x^{4} + 576\,n^{2} + 1080\,n^{2} x - 1296\,n^{2} x^{2} - 729\,n^{2} x^{3} - 216\,n^{3} - 486\,n^{3} x\right)\,P_{n-1}(x) + \left(-32 - 168\,x + 252\,x^{2} + 2943\,x^{3} + 5751\,x^{4} + 3645\,x^{5} + 729\,x^{6} + 48\,n + 864\,n x + 4158\,n x^{2} + 7290\,n x^{3} + 4131\,n x^{4} + 729\,n x^{5} + 144\,n^{2} + 864\,n^{2} x + 1458\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 9 | 6 | 15 | |||||
| 3 | 31 | 60 | 18 | 109 | ||||
| 4 | 101 | 336 | 288 | 54 | 779 | |||
| 5 | 305 | 1452 | 2214 | 1188 | 162 | 5321 | ||
| 6 | 885 | 5502 | 12240 | 11718 | 4536 | 486 | 35367 | |
| 7 | 2495 | 19176 | 56412 | 79488 | 54756 | 16524 | 1458 | 230309 |
Compute more
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Technical details
- Node id
n-635dafcb5fe8fafe· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-5a963a91e97334a6
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: 31, 60, 18; 4: 101, 336, 288, 54; 5: 305, 1452, 2214, 1188, 162; 6: 885, 5502, 12240, 11718, 4536, 486; 7: 2495, 19176, 56412, 79488, 54756, 16524, 1458; 8: 6889, 63120, 231552, 434484, 441612, 236196, 58320, 4374; 9: 18721, 199272, 876996, 2063448, 2802600, 2211300, 963738, 201204, 13122;
%F FormulaThe row polynomials satisfy -(72*n-168-(162*n^3-270*n^2-576*n+1026)*x-(648*n^2-2160*n+1566)*x^2-(243*n^2-486*n-162)*x^3-(243*n-486)*x^4)*P(n,x) = -(216*n^3-576*n^2+96*n+168+(486*n^3-1080*n^2-774*n+1026)*x+(1296*n^2-2916*n+1566)*x^2+(729*n^2-162*n-162)*x^3+(729*n-486)*x^4)*P(n-1,x) + (144*n^2+48*n-32+(864*n^2+864*n-168)*x+(1458*n^2+4158*n+252)*x^2+(486*n^2+7290*n+2943)*x^3+(4131*n+5751)*x^4+(729*n+3645)*x^5+729*x^6)*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[Together[(-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*P[n - 1, x] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*x^4)]];
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] = (-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*v[i - 1] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*x^4)); 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 Laguerre’s theorem on differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] 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…?
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.