Real-rooted atlas

A111125 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(8 + 45\,x + 90\,x^{2} + 77\,x^{3} + 24\,x^{4} - 2\,n - 16\,n x - 38\,n x^{2} - 36\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 31\,x + 75\,x^{2} + 89\,x^{3} + 52\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2459
311291959
42911815971377
5764198617822652403
6199138538355283362298915313
7521438015270282662930816143369197579
8136413441565261315401829651521457004513775621801
93571403571987915575389739451085270753430297980514093962291
10934911918667302922097454649416650197560440703601650124855219186125248833
11244763474552212755832587420499416345128914024219532091120167514575168993716035160892667
1264079100238571073353015365385059809167525785235080713235050375164122929762221922119180326722791025253337
131677612867064223994041057914603364301207589789941245577886149838557613113943958143952653406586998618298699730816533202691
14439204814169369490284361601288127998175032548584066117457542860473068690584210977049244950394179804114999056773481148263722004541631405489
15114985122979309212736331120930766847159379581334687152428271840256455355018845604044260252445849607367415302991868501354665209694341397990291138907099265287027659
Technical details
Node id
n-9ea64980c4dfa120 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ea3393d661da5e9e

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:    4,      5;
   3:   11,     29,     19;
   4:   29,    118,    159,      71;
   5:   76,    419,    861,     782,     265;
   6:  199,   1385,   3835,    5283,    3622,     989;
   7:  521,   4380,  15270,   28266,   29308,   16143,    3691;
   8: 1364,  13441,  56526,  131540,  182965,  152145,   70045,   13775;
   9: 3571,  40357, 198791,  557538,  973945, 1085270,  753430,  297980,   51409;
  10: 9349, 119186, 673029, 2209745, 4649416, 6501975, 6044070, 3601650, 1248552, 191861;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = -(2*n-8+(16*n-45)*x+(38*n-90)*x^2+(36*n-77)*x^3+(12*n-24)*x^4)*P(n-1,x) + (5+31*x+75*x^2+89*x^3+52*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[(-(2*n - 8 + (16*n - 45)*x + (38*n - 90)*x^2 + (36*n - 77)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*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] = (-(2*n - 8 + (16*n - 45)*x + (38*n - 90)*x^2 + (36*n - 77)*x^3 + (12*n - 24)*x^4)*v[i - 1] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*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