A207607 → reverse → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(8 + 32\,x + 48\,x^{2} + 32\,x^{3} + 8\,x^{4} - 16\,n - 64\,n x - 96\,n x^{2} - 64\,n x^{3} - 16\,n x^{4} + 8\,n^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n}(x) = \left(8 + 74\,x + 260\,x^{2} + 490\,x^{3} + 560\,x^{4} + 398\,x^{5} + 164\,x^{6} + 30\,x^{7} - 16\,n - 142\,n x - 490\,n x^{2} - 905\,n x^{3} - 1000\,n x^{4} - 676\,n x^{5} - 262\,n x^{6} - 45\,n x^{7} + 8\,n^{2} + 68\,n^{2} x + 230\,n^{2} x^{2} + 415\,n^{2} x^{3} + 440\,n^{2} x^{4} + 278\,n^{2} x^{5} + 98\,n^{2} x^{6} + 15\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(4\,x + 46\,x^{2} + 224\,x^{3} + 576\,x^{4} + 844\,x^{5} + 710\,x^{6} + 320\,x^{7} + 60\,x^{8} - 4\,n x - 46\,n x^{2} - 200\,n x^{3} - 450\,n x^{4} - 580\,n x^{5} - 434\,n x^{6} - 176\,n x^{7} - 30\,n x^{8}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 30\,x^{4} + 105\,x^{5} + 180\,x^{6} + 165\,x^{7} + 78\,x^{8} + 15\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 2 | 3 | 5 | |||||||||||||
| 2 | 2 | 9 | 8 | 19 | ||||||||||||
| 3 | 2 | 15 | 33 | 21 | 71 | |||||||||||
| 4 | 2 | 21 | 76 | 111 | 55 | 265 | ||||||||||
| 5 | 2 | 27 | 137 | 324 | 355 | 144 | 989 | |||||||||
| 6 | 2 | 33 | 216 | 714 | 1251 | 1098 | 377 | 3691 | ||||||||
| 7 | 2 | 39 | 313 | 1335 | 3256 | 4527 | 3316 | 987 | 13775 | |||||||
| 8 | 2 | 45 | 428 | 2241 | 7045 | 13581 | 15646 | 9837 | 2584 | 51409 | ||||||
| 9 | 2 | 51 | 561 | 3486 | 13455 | 33381 | 53133 | 52248 | 28779 | 6765 | 191861 | |||||
| 10 | 2 | 57 | 712 | 5124 | 23485 | 71505 | 146231 | 198066 | 169877 | 83265 | 17711 | 716035 | ||||
| 11 | 2 | 63 | 881 | 7209 | 38296 | 138474 | 347291 | 603378 | 710942 | 540648 | 238727 | 46368 | 2672279 | |||
| 12 | 2 | 69 | 1068 | 9795 | 59211 | 248238 | 739228 | 1573746 | 2374845 | 2475408 | 1690794 | 679284 | 121393 | 9973081 | ||
| 13 | 2 | 75 | 1273 | 12936 | 87715 | 418662 | 1445646 | 3652956 | 6748792 | 8996565 | 8406076 | 5211018 | 1920518 | 317811 | 37220045 | |
| 14 | 2 | 81 | 1496 | 16686 | 125455 | 672012 | 2642421 | 7741656 | 16968432 | 27669195 | 33020926 | 27953838 | 15862778 | 5400081 | 832040 | 138907099 |
Compute more
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Technical details
- Node id
n-5eae171d920f870a· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-2b83e2879aa49b31
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: 2, 3; 2: 2, 9, 8; 3: 2, 15, 33, 21; 4: 2, 21, 76, 111, 55; 5: 2, 27, 137, 324, 355, 144; 6: 2, 33, 216, 714, 1251, 1098, 377; 7: 2, 39, 313, 1335, 3256, 4527, 3316, 987; 8: 2, 45, 428, 2241, 7045, 13581, 15646, 9837, 2584; 9: 2, 51, 561, 3486, 13455, 33381, 53133, 52248, 28779, 6765;
%F FormulaThe row polynomials satisfy (8*n^2-16*n+8+(32*n^2-64*n+32)*x+(48*n^2-96*n+48)*x^2+(32*n^2-64*n+32)*x^3+(8*n^2-16*n+8)*x^4)*P(n,x) = (8*n^2-16*n+8+(68*n^2-142*n+74)*x+(230*n^2-490*n+260)*x^2+(415*n^2-905*n+490)*x^3+(440*n^2-1000*n+560)*x^4+(278*n^2-676*n+398)*x^5+(98*n^2-262*n+164)*x^6+(15*n^2-45*n+30)*x^7)*P(n-1,x) - ((4*n-4)*x+(46*n-46)*x^2+(200*n-224)*x^3+(450*n-576)*x^4+(580*n-844)*x^5+(434*n-710)*x^6+(176*n-320)*x^7+(30*n-60)*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 >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 8 + (68*n^2 - 142*n + 74)*x + (230*n^2 - 490*n + 260)*x^2 + (415*n^2 - 905*n + 490)*x^3 + (440*n^2 - 1000*n + 560)*x^4 + (278*n^2 - 676*n + 398)*x^5 + (98*n^2 - 262*n + 164)*x^6 + (15*n^2 - 45*n + 30)*x^7)*P[n - 1, x] - ((4*n - 4)*x + (46*n - 46)*x^2 + (200*n - 224)*x^3 + (450*n - 576)*x^4 + (580*n - 844)*x^5 + (434*n - 710)*x^6 + (176*n - 320)*x^7 + (30*n - 60)*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 - 16*n + 8 + (32*n^2 - 64*n + 32)*x + (48*n^2 - 96*n + 48)*x^2 + (32*n^2 - 64*n + 32)*x^3 + (8*n^2 - 16*n + 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 8 + (68*n^2 - 142*n + 74)*x + (230*n^2 - 490*n + 260)*x^2 + (415*n^2 - 905*n + 490)*x^3 + (440*n^2 - 1000*n + 560)*x^4 + (278*n^2 - 676*n + 398)*x^5 + (98*n^2 - 262*n + 164)*x^6 + (15*n^2 - 45*n + 30)*x^7)*v[i - 1] - ((4*n - 4)*x + (46*n - 46)*x^2 + (200*n - 224)*x^3 + (450*n - 576)*x^4 + (580*n - 844)*x^5 + (434*n - 710)*x^6 + (176*n - 320)*x^7 + (30*n - 60)*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 - 16*n + 8 + (32*n^2 - 64*n + 32)*x + (48*n^2 - 96*n + 48)*x^2 + (32*n^2 - 64*n + 32)*x^3 + (8*n^2 - 16*n + 8)*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 v(n,x) jointly generated with A207606; see Formula section.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
Related ?
Candidate equalities ?
- A207607 → basis A007318 → 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