A035607 → reverse → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 - 8\,x - 12\,x^{2} - 8\,x^{3} - 2\,x^{4} + n + 4\,n x + 6\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n}(x) = \left(-2 + 2\,x + 84\,x^{2} + 324\,x^{3} + 566\,x^{4} + 522\,x^{5} + 248\,x^{6} + 48\,x^{7} + n - 7\,n x - 82\,n x^{2} - 278\,n x^{3} - 467\,n x^{4} - 427\,n x^{5} - 204\,n x^{6} - 40\,n x^{7} + 3\,n^{2} x + 20\,n^{2} x^{2} + 58\,n^{2} x^{3} + 92\,n^{2} x^{4} + 83\,n^{2} x^{5} + 40\,n^{2} x^{6} + 8\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(2\,x + 39\,x^{2} + 212\,x^{3} + 546\,x^{4} + 774\,x^{5} + 623\,x^{6} + 268\,x^{7} + 48\,x^{8} - n x - 14\,n x^{2} - 70\,n x^{3} - 176\,n x^{4} - 249\,n x^{5} - 202\,n x^{6} - 88\,n x^{7} - 16\,n x^{8}\right)\,P_{n-1}'(x) + \left(x^{2} + 13\,x^{3} + 58\,x^{4} + 130\,x^{5} + 165\,x^{6} + 121\,x^{7} + 48\,x^{8} + 8\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 3 | 5 | ||||||
| 3 | 2 | 8 | 7 | 17 | |||||
| 4 | 2 | 14 | 28 | 17 | 61 | ||||
| 5 | 2 | 20 | 66 | 88 | 41 | 217 | |||
| 6 | 2 | 26 | 122 | 262 | 262 | 99 | 773 | ||
| 7 | 2 | 32 | 196 | 592 | 940 | 752 | 239 | 2753 | |
| 8 | 2 | 38 | 288 | 1132 | 2508 | 3156 | 2104 | 577 | 9805 |
Compute more
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Technical details
- Node id
n-15c5aa72a34be0a6· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-5db7e795cb165ce2
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 -1)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 2, 3; 3: 2, 8, 7; 4: 2, 14, 28, 17; 5: 2, 20, 66, 88, 41; 6: 2, 26, 122, 262, 262, 99; 7: 2, 32, 196, 592, 940, 752, 239; 8: 2, 38, 288, 1132, 2508, 3156, 2104, 577; 9: 2, 44, 398, 1936, 5560, 9704, 10108, 5776, 1393; 10: 2, 50, 526, 3058, 10852, 24464, 35180, 31252, 15626, 3363;
%F FormulaThe row polynomials satisfy (n-2+(4*n-8)*x+(6*n-12)*x^2+(4*n-8)*x^3+(n-2)*x^4)*P(n,x) = (n-2+(3*n^2-7*n+2)*x+(20*n^2-82*n+84)*x^2+(58*n^2-278*n+324)*x^3+(92*n^2-467*n+566)*x^4+(83*n^2-427*n+522)*x^5+(40*n^2-204*n+248)*x^6+(8*n^2-40*n+48)*x^7)*P(n-1,x) - ((n-2)*x+(14*n-39)*x^2+(70*n-212)*x^3+(176*n-546)*x^4+(249*n-774)*x^5+(202*n-623)*x^6+(88*n-268)*x^7+(16*n-48)*x^8)*d/dx P(n-1,x) + (x^2+13*x^3+58*x^4+130*x^5+165*x^6+121*x^7+48*x^8+8*x^9)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n^2 - 7*n + 2)*x + (20*n^2 - 82*n + 84)*x^2 + (58*n^2 - 278*n + 324)*x^3 + (92*n^2 - 467*n + 566)*x^4 + (83*n^2 - 427*n + 522)*x^5 + (40*n^2 - 204*n + 248)*x^6 + (8*n^2 - 40*n + 48)*x^7)*P[n - 1, x] - ((n - 2)*x + (14*n - 39)*x^2 + (70*n - 212)*x^3 + (176*n - 546)*x^4 + (249*n - 774)*x^5 + (202*n - 623)*x^6 + (88*n - 268)*x^7 + (16*n - 48)*x^8)*D[P[n - 1, x], x] + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*D[P[n - 1, x], {x, 2}])/(n - 2 + (4*n - 8)*x + (6*n - 12)*x^2 + (4*n - 8)*x^3 + (n - 2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (3*n^2 - 7*n + 2)*x + (20*n^2 - 82*n + 84)*x^2 + (58*n^2 - 278*n + 324)*x^3 + (92*n^2 - 467*n + 566)*x^4 + (83*n^2 - 427*n + 522)*x^5 + (40*n^2 - 204*n + 248)*x^6 + (8*n^2 - 40*n + 48)*x^7)*v[i - 1] - ((n - 2)*x + (14*n - 39)*x^2 + (70*n - 212)*x^3 + (176*n - 546)*x^4 + (249*n - 774)*x^5 + (202*n - 623)*x^6 + (88*n - 268)*x^7 + (16*n - 48)*x^8)*deriv(v[i - 1], x) + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (4*n - 8)*x + (6*n - 12)*x^2 + (4*n - 8)*x^3 + (n - 2)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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
reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
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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
Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).?
Related ?
Same fingerprint ?
- A113413 → x ↦ x/(x+1)
- A035607 → x ↦ x+1
- A122542 → reverse → x ↦ x+1
- A122542 → x ↦ x/(x+1)
- A035607 → basis A007318
- A122542 → reverse → basis A007318
- A113413 → reverse → x ↦ x+1
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