Real-rooted atlas

A099097 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(39 + 196\,x + 363\,x^{2} + 294\,x^{3} + 88\,x^{4} - 15\,n - 80\,n x - 159\,n x^{2} - 138\,n x^{3} - 44\,n x^{4}\right)\,P_{n-1}(x) + \left(13 + 87\,x + 227\,x^{2} + 289\,x^{3} + 180\,x^{4} + 44\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2369
310393887
433192372240837
51098432443314415168055
63603474134012583024876957677517
71189137526623717005524544218882060488745983
83927529443057669805681885824217497613928643820807178949
9129701997131344833517257012428961191051281834628010062528241345669086439
1042837741684570522925590462737613631416827341813590481491758887154750415244896664851645
11141481272069723535703120609255405463198934353531149469313416389935761178991440502385136962962886398183439
12467280988117294947246547231500210198833156499875281084413056414861758032142527290409109343232349202937660826752061572760821
13154332135592336376109526240803573410403716362319538542057154099698311764183752014101409374812016175054469093582096240706379523842197696592543948023
1450972431273199461467407916103325999404959294081317133427630843834418598984086561923812094195706281288160860536987576766704516076448544164759282112242697212165702332097133
1516835050452765595565211660043409796300229154464695879541548561253124820813755488345105279310474027586118994877122041140330248455279453648311843804977813344112103272531215330272268854876252562335
Technical details
Node id
n-b7b2c4d89da54564 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ccf1180103c24fbd

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:     3,      6;
   3:    10,     39,      38;
   4:    33,    192,     372,      240;
   5:   109,    843,    2443,     3144,     1516;
   6:   360,   3474,   13401,    25830,    24876,      9576;
   7:  1189,  13752,   66237,   170055,   245442,    188820,     60488;
   8:  3927,  52944,  305766,   980568,  1885824,   2174976,   1392864,    382080;
   9: 12970, 199713, 1344833,  5172570, 12428961,  19105128,  18346280,  10062528,  2413456;
  10: 42837, 741684, 5705229, 25590462, 73761363, 141682734, 181359048, 149175888, 71547504, 15244896;
%F Formula
The row polynomials satisfy (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n,x) = -(15*n-39+(80*n-196)*x+(159*n-363)*x^2+(138*n-294)*x^3+(44*n-88)*x^4)*P(n-1,x) + (13+87*x+227*x^2+289*x^3+180*x^4+44*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[(-(15*n - 39 + (80*n - 196)*x + (159*n - 363)*x^2 + (138*n - 294)*x^3 + (44*n - 88)*x^4)*P[n - 1, x] + (13 + 87*x + 227*x^2 + 289*x^3 + 180*x^4 + 44*x^5)*D[P[n - 1, x], x])/(3*n - 3 + (6*n - 6)*x + (3*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] = (-(15*n - 39 + (80*n - 196)*x + (159*n - 363)*x^2 + (138*n - 294)*x^3 + (44*n - 88)*x^4)*v[i - 1] + (13 + 87*x + 227*x^2 + 289*x^3 + 180*x^4 + 44*x^5)*deriv(v[i - 1], x))/(3*n - 3 + (6*n - 6)*x + (3*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. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741