Real-rooted atlas

A103327 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-4 - 36\,x - 80\,x^{2} - 68\,x^{3} - 20\,x^{4} + 8\,n + 52\,n x + 100\,n x^{2} + 76\,n x^{3} + 20\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 32\,x^{2} - 72\,x^{3} - 64\,x^{4} - 20\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2168080176
36451212168322624
425628161075216640870439168
51024143367577618739221504091136584704
64096696324710401613824293683226787849543688728576
7163843276802703360118784002990080042958848325222409994240130301952

All 15 rows

Technical details
Node id
n-cc6686414670682e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-525575d96ecebcc3

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:      4,       8;
  2:     16,      80,       80;
  3:     64,     512,     1216,       832;
  4:    256,    2816,    10752,     16640,       8704;
  5:   1024,   14336,    75776,    187392,     215040,      91136;
  6:   4096,   69632,   471040,   1613824,    2936832,    2678784,     954368;
  7:  16384,  327680,  2703360,  11878400,   29900800,   42958848,   32522240,    9994240;
  8:  65536, 1507328, 14680064,  78839808,  254607360,  504823808,  598605824,  387383296,  104660992;
  9: 262144, 6815744, 76546048, 486277120, 1921253376, 4884529152, 7973896192, 8045461504, 4546887680, 1096024064;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (8*n-4+(52*n-36)*x+(100*n-80)*x^2+(76*n-68)*x^3+(20*n-20)*x^4)*P(n-1,x) - (4*x+32*x^2+72*x^3+64*x^4+20*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[((8*n - 4 + (52*n - 36)*x + (100*n - 80)*x^2 + (76*n - 68)*x^3 + (20*n - 20)*x^4)*P[n - 1, x] - (4*x + 32*x^2 + 72*x^3 + 64*x^4 + 20*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((8*n - 4 + (52*n - 36)*x + (100*n - 80)*x^2 + (76*n - 68)*x^3 + (20*n - 20)*x^4)*v[i - 1] - (4*x + 32*x^2 + 72*x^3 + 64*x^4 + 20*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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

Same fingerprint ?

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