Real-rooted atlas

A063007 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(5 + 33\,x + 75\,x^{2} + 71\,x^{3} + 24\,x^{4} - n - 11\,n x - 31\,n x^{2} - 33\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 26\,x + 66\,x^{2} + 82\,x^{3} + 50\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2358
313443794
4633215433051232
5321218455566264264116966
616831432548660824606971523525240368
789899186039040588328011220457588202134453468844
8486395800972960391837994514210455144363458135295196182550712992
9265729362292821580384733544001556266602110270401786052808626616018205345748553926
10146256322437477152799408606262272154451530826200830602959602240214663032090718051517019552511131168688
11809745313804902010579298854799165040142717045802907129758441079864420397629607202523150832594779202201600472677166498969924
124504671984488134571957600553673518634512490284774029698790629250391589710061014482826051664504237529137322538598503244191151225159852502416381792
132515959695148375192482421182762737213710001047386276145284745336955256396141931368199115621488868424607814565351824626003316748227716101932159696814345701156937759888297756
14140993361931258302933319579918164199505450430084847934092352595999552131758786861614624997619901083521268705924149951194193651386458086762985043637307740113852105099612442111365435096485571681667171168
157923848253189199514532209582630193314275935858912668029071039332271713880476045789550280323811123336582203264166749996146834118845115913664041596140331542757982472355166848415454767608517108033139123028130334854910778679980422677784
Technical details
Node id
n-39ddaaaec3c44dcf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-84e2bda7face67e7

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 Example
Triangle begins:
   1:       1;
   2:       3,        5;
   3:      13,       44,        37;
   4:      63,      321,       543,       305;
   5:     321,     2184,      5556,      6264,       2641;
   6:    1683,    14325,     48660,     82460,      69715,      23525;
   7:    8989,    91860,    390405,    883280,    1122045,     758820,     213445;
   8:   48639,   580097,   2960391,   8379945,   14210455,   14436345,    8135295,    1961825;
   9:  265729,  3622928,  21580384,  73354400,  155626660,  211027040,  178605280,   86266160,  18205345;
  10: 1462563, 22437477, 152799408, 606262272, 1544515308, 2620083060, 2959602240, 2146630320, 907180515, 170195525;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(n-5+(11*n-33)*x+(31*n-75)*x^2+(33*n-71)*x^3+(12*n-24)*x^4)*P(n-1,x) + (4+26*x+66*x^2+82*x^3+50*x^4+12*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 5 + (11*n - 33)*x + (31*n - 75)*x^2 + (33*n - 71)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 5 + (11*n - 33)*x + (31*n - 75)*x^2 + (33*n - 71)*x^3 + (12*n - 24)*x^4)*v[i - 1] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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

Proof chains ? JSON

References

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