Real-rooted atlas

A207824 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −3/4)coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(26 + 110\,x + 174\,x^{2} + 122\,x^{3} + 32\,x^{4} - 16\,n - 78\,n x - 140\,n x^{2} - 110\,n x^{3} - 32\,n x^{4}\right)\,P_{n-1}(x) + \left(21 + 115\,x + 251\,x^{2} + 273\,x^{3} + 148\,x^{4} + 32\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15611
2245835117
31154185062041243
4551267448633928118913205
52640160283890247184285996930140283
61264992204279916453004412191199938403911490293
7606055156061879206380350046171103361524135912423541615832091
82903762824158120126932918769644308435430329562611230690511041372105168191829
9139127515226230740373902099371203825650354646161463760599841956141685933719379972141786781755
106665999810738904435929451437872700305775137544576334184511510143313012207214247843543842139144661117918981831989
1131938720427356664259852694094776830402303951641039194992004476154939124130732093625078028689101474899262463007758271669860201653024859
1215302760122340097521494436796060573444360165686590570322201049612456762893230475618792376361037530420194841717070709601549851565895957615834079812142255945493
137331992851159698026284640719324377431335976114728020958025103730836164067975769492494293802049645035922418833038822899630147601506104648871055248898864449883922877802622758203300987
143512968824598440320664732200631272302304805792769921121650618721399000448341352657661324741119909098450406551323989408231323791622479133158346510947222142100332272408222562050481389053789260175241771212528885
Technical details
Node id
n-8a6cb57f8c79f2fa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-132406c0e640de7f

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:       5,        6;
  2:      24,       58,       35;
  3:     115,      418,      506,       204;
  4:     551,     2674,     4863,      3928,      1189;
  5:    2640,    16028,    38902,     47184,     28599,      6930;
  6:   12649,    92204,   279916,    453004,    412191,    199938,     40391;
  7:   60605,   515606,  1879206,   3803500,   4617110,   3361524,   1359124,    235416;
  8:  290376,  2824158, 12012693,  29187696,  44308435,  43032956,  26112306,   9051104,  1372105;
  9: 1391275, 15226230, 74037390, 209937120, 382565035, 464616146, 376059984, 195614168, 59337193, 7997214;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(16*n-26+(78*n-110)*x+(140*n-174)*x^2+(110*n-122)*x^3+(32*n-32)*x^4)*P(n-1,x) + (21+115*x+251*x^2+273*x^3+148*x^4+32*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[(-(16*n - 26 + (78*n - 110)*x + (140*n - 174)*x^2 + (110*n - 122)*x^3 + (32*n - 32)*x^4)*P[n - 1, x] + (21 + 115*x + 251*x^2 + 273*x^3 + 148*x^4 + 32*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = (-(16*n - 26 + (78*n - 110)*x + (140*n - 174)*x^2 + (110*n - 122)*x^3 + (32*n - 32)*x^4)*v[i - 1] + (21 + 115*x + 251*x^2 + 273*x^3 + 148*x^4 + 32*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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

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