Real-rooted atlas

A046716 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 26\,x^{2} + 24\,x^{3} + 8\,x^{4} - n - 6\,n x - 13\,n x^{2} - 12\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}(x) + \left(-x - 2\,x^{2} - x^{3} + n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
35151131
4167310953251
5654049299343452677
632625918158126999752294736473
71957190557676416361719436612182531411608995
813700158425780981212622634501033333584177388740043712037343
91096011470544859637028584460591055397777770863562686294694045927921274604233
10986410150820591021481884021048301013528682169567112118821342321335742506549619518998165157096834061
119864101169436727130610075759485960341772255720136079645317508127090274886450761830693900795113656878431882741659204855747079
1210850511220698380571790603316592713211639319182987654766992507737131239501617515985967598571357999220054765935326291258026278977393103975576531236082275
13130206134527322698100262268037811152256437476259530543424621651207045110433309711193611492308601075825289547841600240281547206688206313927256556378761244578899919305929227906252165629
14169267974863876271276554090032507962263257379092411153120793106023629265259030228442766949174941469600481467585191374632483855818410117471744661271396099167462596728379193229170553263412980224109221773478638782841905089
1523697516480558823817766796769370233874247864495623023523229265971212178184559099392788215878069401676734329729754769735566344857908068273772795899110218606734686871775804942571015791093483184471949903691484903717083010261603120939234755353441413540445323
Technical details
Node id
n-fa7b0940b6c511f6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-87a116103433ab13

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:      2,        3;
   3:      5,       15,        11;
   4:     16,       73,       109,        53;
   5:     65,      404,       929,       934,        345;
   6:    326,     2591,      8158,     12699,       9752,       2947;
   7:   1957,    19055,     76764,    163617,     194366,     121825,      31411;
   8:  13700,   158425,    780981,   2126226,    3450103,    3333584,    1773887,     400437;
   9: 109601,  1470544,   8596370,  28584460,   59105539,   77777708,   63562686,   29469404,   5927921;
  10: 986410, 15082059, 102148188, 402104830, 1013528682, 1695671121, 1882134232, 1335742506, 549619518, 99816515;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(6*n-12)*x+(13*n-26)*x^2+(12*n-24)*x^3+(4*n-8)*x^4)*P(n-2,x) + (n+(4*n-1)*x+(5*n-2)*x^2+(2*n-1)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (6*n - 12)*x + (13*n - 26)*x^2 + (12*n - 24)*x^3 + (4*n - 8)*x^4)*P[n - 2, x] + (n + (4*n - 1)*x + (5*n - 2)*x^2 + (2*n - 1)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (6*n - 12)*x + (13*n - 26)*x^2 + (12*n - 24)*x^3 + (4*n - 8)*x^4)*v[i - 2] + (n + (4*n - 1)*x + (5*n - 2)*x^2 + (2*n - 1)*x^3)*v[i - 1])/(1 + 2*x + 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. [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