Real-rooted atlas

A091044 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-6\,x - 22\,x^{2} - 26\,x^{3} - 10\,x^{4} + 4\,n + 26\,n x + 50\,n x^{2} + 38\,n x^{3} + 10\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 16\,x^{2} - 36\,x^{3} - 32\,x^{4} - 10\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
21696128240
364576160013443584
4256307213056230401408053504
510241536088064239616307200147456798720
6409673728532480196608038952963907584154419211923456
71638434406429982721400832037765120585400324815257616171008177995776
865536157286416056320908328963106406406558842888327659525799936001693450242657157120
92621447077888828375045505024002284584960612709171210597433344113765253126866862080177340416039666581504
10104857631457280415236096317089382415487467520504700600321109445836801621811527681506614968328021606400018571329536592150790144
1141943041384120322034237440175531622409867519590437892181196810128656957441882026147840237926088704019467778129929270082928641944814878728839747338240
12167772166039797769781116928941201817605986949529602649055297536835072504627218874971979776303262439833603374334345216024658782978048106185684418562036636581888131961544769536
136710886426172456964630511616049163953766434916741939201749971002982463612760621056169645839482880331598788558848468775883243520465487079145472307231224496128120737234944000213279351767041969948758900736
1426843545611274289152216358977536251255586816019709873356800110393007538176454868443594752139955160062361632285336272896005552289444003840700203901177036862748763524956163775183106080768136426146496512022334903681024029407795426099200
Technical details
Node id
n-8472e96e8a810be8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-05007905755e7549

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,      12;
  2:     16,      96,      128;
  3:     64,     576,     1600,      1344;
  4:    256,    3072,    13056,     23040,      14080;
  5:   1024,   15360,    88064,    239616,     307200,     147456;
  6:   4096,   73728,   532480,   1966080,    3895296,    3907584,     1544192;
  7:  16384,  344064,  2998272,  14008320,   37765120,   58540032,    48152576,    16171008;
  8:  65536, 1572864, 16056320,  90832896,  310640640,  655884288,   832765952,   579993600,  169345024;
  9: 262144, 7077888, 82837504, 550502400, 2284584960, 6127091712, 10597433344, 11376525312, 6866862080, 1773404160;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n+(26*n-6)*x+(50*n-22)*x^2+(38*n-26)*x^3+(10*n-10)*x^4)*P(n-1,x) - (2*x+16*x^2+36*x^3+32*x^4+10*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 + (26*n - 6)*x + (50*n - 22)*x^2 + (38*n - 26)*x^3 + (10*n - 10)*x^4)*P[n - 1, x] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*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 + (26*n - 6)*x + (50*n - 22)*x^2 + (38*n - 26)*x^3 + (10*n - 10)*x^4)*v[i - 1] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*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