Real-rooted atlas

A110441 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(12 + 53\,x + 87\,x^{2} + 63\,x^{3} + 17\,x^{4} - 4\,n - 27\,n x - 59\,n x^{2} - 53\,n x^{3} - 17\,n x^{4}\right)\,P_{n-1}(x) + \left(8 + 48\,x + 113\,x^{2} + 131\,x^{3} + 75\,x^{4} + 17\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
214362373
348188244105585
4164864169814764794681
556037089776128328388218537449
6191215248504648873687454458129967299593
7652860896242592535080705840556932243412454652396745

All 15 rows

Technical details
Node id
n-bca827b180da6bd4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ca1f2c2a839ffbec

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,      5;
  2:    14,     36,      23;
  3:    48,    188,     244,      105;
  4:   164,    864,    1698,     1476,      479;
  5:   560,   3708,    9776,    12832,     8388,     2185;
  6:  1912,  15248,   50464,    88736,    87454,    45812,     9967;
  7:  6528,  60896,  242592,   535080,   705840,   556932,   243412,    45465;
  8: 22288, 238080, 1109104,  2943456,  4867980,  5138016,  3380218,  1267428,  207391;
  9: 76096, 915856, 4884992, 15157024, 30152016, 39884580, 35084720, 19792360, 6498020, 946025;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(4*n-12+(27*n-53)*x+(59*n-87)*x^2+(53*n-63)*x^3+(17*n-17)*x^4)*P(n-1,x) + (8+48*x+113*x^2+131*x^3+75*x^4+17*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[(-(4*n - 12 + (27*n - 53)*x + (59*n - 87)*x^2 + (53*n - 63)*x^3 + (17*n - 17)*x^4)*P[n - 1, x] + (8 + 48*x + 113*x^2 + 131*x^3 + 75*x^4 + 17*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*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] = (-(4*n - 12 + (27*n - 53)*x + (59*n - 87)*x^2 + (53*n - 63)*x^3 + (17*n - 17)*x^4)*v[i - 1] + (8 + 48*x + 113*x^2 + 131*x^3 + 75*x^4 + 17*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023