Real-rooted atlas

A114687 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4 + 8\,x + 4\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(3 - 4\,x - 62\,x^{2} - 100\,x^{3} - 45\,x^{4} + 15\,n + 106\,n x + 212\,n x^{2} + 166\,n x^{3} + 45\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 17\,x - 90\,x^{2} - 178\,x^{3} - 149\,x^{4} - 45\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131114
21178131220
345471157516613752
4197272413646291242210167792
5903155151035903342305176353051511277024
6427987858733185317142074655459017298433571124815296
7207934965594972653269951158552071515749134115538158363009881493888640
810304928058643278332421422402485420693421222457203197584796266107524093244996110017478912
951885915864075211846860161897190077880795302440268245849679314524631857359164541397129914004694451206321679872
1026467238977887013488525751180395540066528360150251897026500647626461510111396228948012238716361357734951980702129440330514303570238464
11136488695086405838491584915837354284255412281440502404255505190747835024167016259429690130241701373222652334736036451513160097049143327161829666190725225916416
127103937328849902845298380029858131669222042391807746752162114968860078983590059660207934262003160390931511816835511033168564380440111951253546223789351481388507115641523811929992986316800
1337269351916382070939328233190194396596230859432212619302149185546842216593778934606953932241022855402986854900595584175779079684134568333105821063089727708217647489054146380498601349795579208810459351141377805181870080
141968801519931250823302021467911645266719929676202387944036227751533326944130982727542252615278125888602982300344693493526889173011390115834998030142050067743159025592154133662155715241522499680582490616469909439274527412454801769554551893149876351713280
Technical details
Node id
n-db8916c6505a6d8b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e9cd41522307e935

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:      3,       11;
  2:     11,       78,       131;
  3:     45,      471,      1575,       1661;
  4:    197,     2724,     13646,      29124,      22101;
  5:    903,    15515,    103590,     334230,     517635,      305151;
  6:   4279,    87858,    733185,    3171420,    7465545,     9017298,     4335711;
  7:  20793,   496559,   4972653,   26995115,   85520715,   157491341,   155381583,    63009881;
  8: 103049,  2805864,  32783324,  214224024,  854206934,  2122245720,  3197584796,  2661075240,   932449961;
  9: 518859, 15864075, 211846860, 1618971900, 7788079530, 24402682458, 49679314524, 63185735916, 45413971299, 14004694451;
%F Formula
The row polynomials satisfy (2*n+4+(4*n+8)*x+(2*n+4)*x^2)*P(n,x) = (15*n+3+(106*n-4)*x+(212*n-62)*x^2+(166*n-100)*x^3+(45*n-45)*x^4)*P(n-1,x) - (1+17*x+90*x^2+178*x^3+149*x^4+45*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[((15*n + 3 + (106*n - 4)*x + (212*n - 62)*x^2 + (166*n - 100)*x^3 + (45*n - 45)*x^4)*P[n - 1, x] - (1 + 17*x + 90*x^2 + 178*x^3 + 149*x^4 + 45*x^5)*D[P[n - 1, x], x])/(2*n + 4 + (4*n + 8)*x + (2*n + 4)*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] = ((15*n + 3 + (106*n - 4)*x + (212*n - 62)*x^2 + (166*n - 100)*x^3 + (45*n - 45)*x^4)*v[i - 1] - (1 + 17*x + 90*x^2 + 178*x^3 + 149*x^4 + 45*x^5)*deriv(v[i - 1], x))/(2*n + 4 + (4*n + 8)*x + (2*n + 4)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023