Real-rooted atlas

A099097 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-6 - 12\,x - 6\,x^{2} + 6\,n + 12\,n x + 6\,n x^{2}\right)\,P_{n}(x) = \left(52 + 306\,x + 642\,x^{2} + 574\,x^{3} + 186\,x^{4} - 17\,n - 108\,n x - 258\,n x^{2} - 260\,n x^{3} - 93\,n x^{4}\right)\,P_{n-1}(x) + \left(13 + 109\,x + 342\,x^{2} + 502\,x^{3} + 349\,x^{4} + 93\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
23912
3105884152
4332858197831920
51091250536810230729924256
63605148294128391611955668040306432
7118920370145263551908117823513400106342573871232
8392778399670203318009990453331542215714593689591243348906240
9129702956662946510167661945957854613538314219210948215563755855114668617844736
10428371097835124961048291328035340772210035001261898195040230645167216335128255137693117805386752
111414814026584515369113906440081941925538661516578415638463942253333352882691259686916930294704478926780398607398912
124672801462226420786594417719554241006412184039988698048113424302592229655913552325291712832306971431560173696071464446447181601245731880960
131543321526648208232658987795531172497990766392260979906647480960939001817519152392321793969031140491203982123437079335658176714675298041617026684915737641762816
145097243188375733321153791433443272590237332551833121204671342345827416579801298874152034027595679717277434059244309794913077852708237895497578558178524986643993879466556121198817556201472
151683505066984049012368550480140480248220109643810783062208139975022645853068122085699096982824212418017416470400952327365062567333035459496583526324769660412412969485962179277008974290361637098056362511711808520192
Technical details
Node id
n-c69ac1d7fd0ba79f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-110346e0d710e6d1

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,       9;
   3:    10,      58,       84;
   4:    33,     285,      819,      783;
   5:   109,    1250,     5368,    10230,      7299;
   6:   360,    5148,    29412,    83916,    119556,      68040;
   7:  1189,   20370,   145263,   551908,   1178235,    1340010,     634257;
   8:  3927,   78399,   670203,  3180099,   9045333,   15422157,   14593689,    5912433;
   9: 12970,  295666,  2946510, 16766194,  59578546,  135383142,  192109482,  155637558,   55114668;
  10: 42837, 1097835, 12496104, 82913280, 353407722, 1003500126, 1898195040, 2306451672, 1633512825, 513769311;
%F Formula
The row polynomials satisfy (6*n-6+(12*n-12)*x+(6*n-6)*x^2)*P(n,x) = -(17*n-52+(108*n-306)*x+(258*n-642)*x^2+(260*n-574)*x^3+(93*n-186)*x^4)*P(n-1,x) + (13+109*x+342*x^2+502*x^3+349*x^4+93*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[(-(17*n - 52 + (108*n - 306)*x + (258*n - 642)*x^2 + (260*n - 574)*x^3 + (93*n - 186)*x^4)*P[n - 1, x] + (13 + 109*x + 342*x^2 + 502*x^3 + 349*x^4 + 93*x^5)*D[P[n - 1, x], x])/(6*n - 6 + (12*n - 12)*x + (6*n - 6)*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] = (-(17*n - 52 + (108*n - 306)*x + (258*n - 642)*x^2 + (260*n - 574)*x^3 + (93*n - 186)*x^4)*v[i - 1] + (13 + 109*x + 342*x^2 + 502*x^3 + 349*x^4 + 93*x^5)*deriv(v[i - 1], x))/(6*n - 6 + (12*n - 12)*x + (6*n - 6)*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