A207606 → reverse → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(18 + 72\,x + 108\,x^{2} + 72\,x^{3} + 18\,x^{4} + 24\,n + 96\,n x + 144\,n x^{2} + 96\,n x^{3} + 24\,n x^{4} + 8\,n^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n}(x) = \left(18 + 141\,x + 441\,x^{2} + 714\,x^{3} + 636\,x^{4} + 297\,x^{5} + 57\,x^{6} + 24\,n + 196\,n x + 639\,n x^{2} + 1091\,n x^{3} + 1054\,n x^{4} + 570\,n x^{5} + 155\,n x^{6} + 15\,n x^{7} + 8\,n^{2} + 68\,n^{2} x + 230\,n^{2} x^{2} + 415\,n^{2} x^{3} + 440\,n^{2} x^{4} + 278\,n^{2} x^{5} + 98\,n^{2} x^{6} + 15\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(-6\,x - 69\,x^{2} - 273\,x^{3} - 522\,x^{4} - 528\,x^{5} - 273\,x^{6} - 57\,x^{7} - 4\,n x - 46\,n x^{2} - 200\,n x^{3} - 450\,n x^{4} - 580\,n x^{5} - 434\,n x^{6} - 176\,n x^{7} - 30\,n x^{8}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 30\,x^{4} + 105\,x^{5} + 180\,x^{6} + 165\,x^{7} + 78\,x^{8} + 15\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 2 + 11\,x + 13\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2 | 11 | 13 | 26 | |||||||
| 1 | 2 | 17 | 44 | 34 | 97 | ||||||
| 2 | 2 | 23 | 93 | 155 | 89 | 362 | |||||
| 3 | 2 | 29 | 160 | 417 | 510 | 233 | 1351 | ||||
| 4 | 2 | 35 | 245 | 874 | 1668 | 1608 | 610 | 5042 | |||
| 5 | 2 | 41 | 348 | 1580 | 4130 | 6195 | 4924 | 1597 | 18817 | ||
| 6 | 2 | 47 | 469 | 2589 | 8625 | 17711 | 21841 | 14761 | 4181 | 70226 | |
| 7 | 2 | 53 | 608 | 3955 | 16044 | 42006 | 70844 | 74089 | 43540 | 10946 | 262087 |
Compute more
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Technical details
- Node id
n-91c62c75d8f18841· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-797bb38c8aa87a31
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
- 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, 11, 13; 1: 2, 17, 44, 34; 2: 2, 23, 93, 155, 89; 3: 2, 29, 160, 417, 510, 233; 4: 2, 35, 245, 874, 1668, 1608, 610; 5: 2, 41, 348, 1580, 4130, 6195, 4924, 1597; 6: 2, 47, 469, 2589, 8625, 17711, 21841, 14761, 4181; 7: 2, 53, 608, 3955, 16044, 42006, 70844, 74089, 43540, 10946; 8: 2, 59, 765, 5732, 27440, 87549, 188237, 268910, 243966, 126805, 28657; 9: 2, 65, 940, 7974, 44028, 165914, 434840, 791615, 979852, 784614, 365532, 75025;
%F FormulaThe row polynomials satisfy (8*n^2+24*n+18+(32*n^2+96*n+72)*x+(48*n^2+144*n+108)*x^2+(32*n^2+96*n+72)*x^3+(8*n^2+24*n+18)*x^4)*P(n,x) = (8*n^2+24*n+18+(68*n^2+196*n+141)*x+(230*n^2+639*n+441)*x^2+(415*n^2+1091*n+714)*x^3+(440*n^2+1054*n+636)*x^4+(278*n^2+570*n+297)*x^5+(98*n^2+155*n+57)*x^6+(15*n^2+15*n)*x^7)*P(n-1,x) - ((4*n+6)*x+(46*n+69)*x^2+(200*n+273)*x^3+(450*n+522)*x^4+(580*n+528)*x^5+(434*n+273)*x^6+(176*n+57)*x^7+30*n*x^8)*d/dx P(n-1,x) + (3*x^3+30*x^4+105*x^5+180*x^6+165*x^7+78*x^8+15*x^9)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2 + 11*x + 13*x^2.
%t MathematicaP[0, x_] := 2 + 11*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 24*n + 18 + (68*n^2 + 196*n + 141)*x + (230*n^2 + 639*n + 441)*x^2 + (415*n^2 + 1091*n + 714)*x^3 + (440*n^2 + 1054*n + 636)*x^4 + (278*n^2 + 570*n + 297)*x^5 + (98*n^2 + 155*n + 57)*x^6 + (15*n^2 + 15*n)*x^7)*P[n - 1, x] - ((4*n + 6)*x + (46*n + 69)*x^2 + (200*n + 273)*x^3 + (450*n + 522)*x^4 + (580*n + 528)*x^5 + (434*n + 273)*x^6 + (176*n + 57)*x^7 + 30*n*x^8)*D[P[n - 1, x], x] + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (32*n^2 + 96*n + 72)*x + (48*n^2 + 144*n + 108)*x^2 + (32*n^2 + 96*n + 72)*x^3 + (8*n^2 + 24*n + 18)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 2 + 11*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 24*n + 18 + (68*n^2 + 196*n + 141)*x + (230*n^2 + 639*n + 441)*x^2 + (415*n^2 + 1091*n + 714)*x^3 + (440*n^2 + 1054*n + 636)*x^4 + (278*n^2 + 570*n + 297)*x^5 + (98*n^2 + 155*n + 57)*x^6 + (15*n^2 + 15*n)*x^7)*v[i - 1] - ((4*n + 6)*x + (46*n + 69)*x^2 + (200*n + 273)*x^3 + (450*n + 522)*x^4 + (580*n + 528)*x^5 + (434*n + 273)*x^6 + (176*n + 57)*x^7 + 30*n*x^8)*deriv(v[i - 1], x) + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (32*n^2 + 96*n + 72)*x + (48*n^2 + 144*n + 108)*x^2 + (32*n^2 + 96*n + 72)*x^3 + (8*n^2 + 24*n + 18)*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 [-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
reversal of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
Related ?
Candidate equalities ?
- A207606 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found
Same fingerprint ?
References
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023