Real-rooted atlas

A085841 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 - 13\,x - 41\,x^{2} - 39\,x^{3} - 12\,x^{4} + 2\,n + 26\,n x + 58\,n x^{2} + 46\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x - 24\,x^{3} - 32\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
255061116
371616375471352
4937230787380492115760
51171510142473668119144287183712
6131222265072038926476998635903985812141504
715192559283679905344604582207358968065358722724963200
817285611852418967921402823052089496990971489147428032285041290992384
919404721774046372924712421825079768272843119611501545889201236592905653673392055808
102155303744091024063213690716298784630040520249709618924280129664795059152809050261508830139540700160
1123733761048920858079354960518333497621818334749874607213517101215471249151428491984459022054637323535794707460920178688
1225950095293039777100839668775996098860070662150300310592811000858586608375148324539550015445365276508825923015002118221523615372879343616
132712051143418671822322184180210526925729129239414649372134707948657249008695632291159229826040517594477530362164426945770268578797170607190639937124762630871408640
1429150222092727123841484379224488566988509458729633352359511203666026423641544897103730662311088501507395940347492038655827625721727674430192638292837264923017157594341221730078939529216
Technical details
Node id
n-42294897d817716e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12ddcff56b5c7647

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:  3,    7;
  2:  5,   50,     61;
  3:  7,  161,    637,     547;
  4:  9,  372,   3078,    7380,     4921;
  5: 11,  715,  10142,   47366,    81191,     44287;
  6: 13, 1222,  26507,  203892,   647699,    863590,    398581;
  7: 15, 1925,  59283,  679905,  3446045,   8220735,   8968065,    3587227;
  8: 17, 2856, 118524, 1896792, 14028230,  52089496,  99097148,   91474280,  32285041;
  9: 19, 4047, 217740, 4637292, 47124218, 250797682, 728431196, 1150154588, 920123659, 290565367;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2*n+1+(26*n-13)*x+(58*n-41)*x^2+(46*n-39)*x^3+(12*n-12)*x^4)*P(n-1,x) + (4*x-24*x^3-32*x^4-12*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[((2*n + 1 + (26*n - 13)*x + (58*n - 41)*x^2 + (46*n - 39)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] + (4*x - 24*x^3 - 32*x^4 - 12*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((2*n + 1 + (26*n - 13)*x + (58*n - 41)*x^2 + (46*n - 39)*x^3 + (12*n - 12)*x^4)*v[i - 1] + (4*x - 24*x^3 - 32*x^4 - 12*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*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 - 1, v[i])))};
T(9)

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. [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