A249250 → x ↦ x+1 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(41 + 299\,x + 887\,x^{2} + 1377\,x^{3} + 1184\,x^{4} + 536\,x^{5} + 100\,x^{6} - 68\,n - 479\,n x - 1387\,n x^{2} - 2117\,n x^{3} - 1799\,n x^{4} - 808\,n x^{5} - 150\,n x^{6} + 27\,n^{2} + 180\,n^{2} x + 500\,n^{2} x^{2} + 740\,n^{2} x^{3} + 615\,n^{2} x^{4} + 272\,n^{2} x^{5} + 50\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(66 + 561\,x + 2014\,x^{2} + 3966\,x^{3} + 4634\,x^{4} + 3217\,x^{5} + 1230\,x^{6} + 200\,x^{7} - 36\,n - 300\,n x - 1059\,n x^{2} - 2056\,n x^{3} - 2374\,n x^{4} - 1632\,n x^{5} - 619\,n x^{6} - 100\,n x^{7}\right)\,P_{n-1}'(x) + \left(12 + 120\,x + 517\,x^{2} + 1255\,x^{3} + 1880\,x^{4} + 1782\,x^{5} + 1045\,x^{6} + 347\,x^{7} + 50\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 7\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 7 | 11 | ||||||
| 2 | 10 | 32 | 25 | 67 | |||||
| 3 | 28 | 132 | 204 | 103 | 467 | ||||
| 4 | 76 | 472 | 1086 | 1096 | 409 | 3139 | |||
| 5 | 208 | 1604 | 4900 | 7408 | 5540 | 1639 | 21299 | ||
| 6 | 568 | 5232 | 19920 | 40112 | 45042 | 26736 | 6553 | 144163 | |
| 7 | 1552 | 16624 | 75792 | 190616 | 285548 | 254748 | 125308 | 26215 | 976403 |
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Technical details
- Node id
n-6ea4d72dc1454e17· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-9e00e04d10cd46cf
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: 4, 7; 2: 10, 32, 25; 3: 28, 132, 204, 103; 4: 76, 472, 1086, 1096, 409; 5: 208, 1604, 4900, 7408, 5540, 1639; 6: 568, 5232, 19920, 40112, 45042, 26736, 6553; 7: 1552, 16624, 75792, 190616, 285548, 254748, 125308, 26215; 8: 4240, 51776, 274960, 829280, 1553380, 1850312, 1368550, 574616, 104857; 9: 11584, 158832, 962784, 3385920, 7612560, 11346012, 11209332, 7078152, 2592084, 419431;
%F FormulaThe row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (27*n^2-68*n+41+(180*n^2-479*n+299)*x+(500*n^2-1387*n+887)*x^2+(740*n^2-2117*n+1377)*x^3+(615*n^2-1799*n+1184)*x^4+(272*n^2-808*n+536)*x^5+(50*n^2-150*n+100)*x^6)*P(n-1,x) - (36*n-66+(300*n-561)*x+(1059*n-2014)*x^2+(2056*n-3966)*x^3+(2374*n-4634)*x^4+(1632*n-3217)*x^5+(619*n-1230)*x^6+(100*n-200)*x^7)*d/dx P(n-1,x) + (12+120*x+517*x^2+1255*x^3+1880*x^4+1782*x^5+1045*x^6+347*x^7+50*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 7*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4 + 7*x;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 68*n + 41 + (180*n^2 - 479*n + 299)*x + (500*n^2 - 1387*n + 887)*x^2 + (740*n^2 - 2117*n + 1377)*x^3 + (615*n^2 - 1799*n + 1184)*x^4 + (272*n^2 - 808*n + 536)*x^5 + (50*n^2 - 150*n + 100)*x^6)*P[n - 1, x] - (36*n - 66 + (300*n - 561)*x + (1059*n - 2014)*x^2 + (2056*n - 3966)*x^3 + (2374*n - 4634)*x^4 + (1632*n - 3217)*x^5 + (619*n - 1230)*x^6 + (100*n - 200)*x^7)*D[P[n - 1, x], x] + (12 + 120*x + 517*x^2 + 1255*x^3 + 1880*x^4 + 1782*x^5 + 1045*x^6 + 347*x^7 + 50*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 7*x; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^2 - 68*n + 41 + (180*n^2 - 479*n + 299)*x + (500*n^2 - 1387*n + 887)*x^2 + (740*n^2 - 2117*n + 1377)*x^3 + (615*n^2 - 1799*n + 1184)*x^4 + (272*n^2 - 808*n + 536)*x^5 + (50*n^2 - 150*n + 100)*x^6)*v[i - 1] - (36*n - 66 + (300*n - 561)*x + (1059*n - 2014)*x^2 + (2056*n - 3966)*x^3 + (2374*n - 4634)*x^4 + (1632*n - 3217)*x^5 + (619*n - 1230)*x^6 + (100*n - 200)*x^7)*deriv(v[i - 1], x) + (12 + 120*x + 517*x^2 + 1255*x^3 + 1880*x^4 + 1782*x^5 + 1045*x^6 + 347*x^7 + 50*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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, 0]
- 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.?
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