Real-rooted atlas

A308737 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 21\,x^{2} + 19\,x^{3} + 6\,x^{4} - 2\,n x - 6\,n x^{2} - 6\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
21101627
31206864153
4138204392256891
51725361568206410245265
61138132052001032010272409631347
7126831321552041200616324921616384187353
81526725243288143920287392344512229504655361121931
9110401649611513645987211486721836800183603210488322621446725025
1012066370082955841378272413280082629121101465694394884719104104857640330467
11141168210073824039355201377062433043584550656006292608047191040209725444194304241923753
12182141804441803560108187204326854412114009624226329634607232034605824023069798492276736167772161451365371
131164083934804327136288437601297866244152952329689856001661071104207629312018455613441107320832402657280671088648707660785
141327948522801022590474984624374907104134962713635989324807197755136107965045761199600230495967272965234544640174483865626843545652244370387
Technical details
Node id
n-75137e04e9c9ba8a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3de95bbd8b3c6beb

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,    4;
  2: 1,   10,    16;
  3: 1,   20,    68,     64;
  4: 1,   38,   204,    392,    256;
  5: 1,   72,   536,   1568,   2064,    1024;
  6: 1,  138,  1320,   5200,  10320,   10272,    4096;
  7: 1,  268,  3132,  15520,  41200,   61632,   49216,   16384;
  8: 1,  526,  7252,  43288, 143920,  287392,  344512,  229504,   65536;
  9: 1, 1040, 16496, 115136, 459872, 1148672, 1836800, 1836032, 1048832, 262144;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(6*n-21)*x^2-(6*n-19)*x^3-(2*n-6)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*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[((1 - (2*n - 9)*x - (6*n - 21)*x^2 - (6*n - 19)*x^3 - (2*n - 6)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = ((1 - (2*n - 9)*x - (6*n - 21)*x^2 - (6*n - 19)*x^3 - (2*n - 6)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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. [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