Real-rooted atlas

A091044 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-3\,x - 10\,x^{2} - 11\,x^{3} - 4\,x^{4} + 4\,n + 17\,n x + 26\,n x^{2} + 17\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
216483599
364288424204980
425615363408331211899701
510247680227843340824204693096030
640963686413696026880029385616963240391950599
71638417203276800018892802765440240825611551522354169409960
8655367864324100096121282562226048025956864187762887702944137210593149001
926214435389442110259272941568161043456235501056228080896141061248505509327997214922080050
1010485761572864010557849641759539210777231361896136704230306944019067735041029794928327591504466111799127651499
1141943046920601651642368023003136006795362304139779440642042805452821210009600153317398407348074336210144280827166986090354434940
1216777216301989888247988224012284067840408777523209626635468816450215936020551222886418628394752011947596288051464422320133677447841583407981894416697901
13671088641308622848117272739846394635878423669899264062803161907212288114032641795083927552195778110873615743975270409073817267203549094387208443865782092287780268853812544070
1426843545656371445765474405580832584708915213279942410243920036364288864255757516814456937775104184371258408961783876453171212889628121600674445215232024157468412805301793142405378926017587643708742799
Technical details
Node id
n-2f082b59c5b072d9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-719d0cceaf47e4d3

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 Example
Triangle begins:
  0:      1;
  1:      4,       6;
  2:     16,      48,       35;
  3:     64,     288,      424,      204;
  4:    256,    1536,     3408,     3312,      1189;
  5:   1024,    7680,    22784,    33408,     24204,      6930;
  6:   4096,   36864,   136960,   268800,    293856,    169632,     40391;
  7:  16384,  172032,   768000,  1889280,   2765440,   2408256,   1155152,    235416;
  8:  65536,  786432,  4100096, 12128256,  22260480,  25956864,  18776288,   7702944,  1372105;
  9: 262144, 3538944, 21102592, 72941568, 161043456, 235501056, 228080896, 141061248, 50550932, 7997214;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n+(17*n-3)*x+(26*n-10)*x^2+(17*n-11)*x^3+(4*n-4)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (17*n - 3)*x + (26*n - 10)*x^2 + (17*n - 11)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + (17*n - 3)*x + (26*n - 10)*x^2 + (17*n - 11)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023