Real-rooted atlas

A103327 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-4 - 31\,x - 66\,x^{2} - 55\,x^{3} - 16\,x^{4} + 8\,n + 46\,n x + 84\,n x^{2} + 62\,n x^{3} + 16\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 28\,x^{2} - 60\,x^{3} - 52\,x^{4} - 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14711
2166861145
3644328885471931
42562368772810456492125729
51024120325401611521611625244287342827
6409658368334080979712154761612455643985814568017
71638427443219107847151360155014401942147213003696358722760866795

All 15 rows

Technical details
Node id
n-bd974cc73ca079ba · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-61c8a024c8013bac

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,       7;
  2:     16,      68,       61;
  3:     64,     432,      888,       547;
  4:    256,    2368,     7728,     10456,      4921;
  5:   1024,   12032,    54016,    115216,    116252,      44287;
  6:   4096,   58368,   334080,    979712,   1547616,    1245564,     398581;
  7:  16384,  274432,  1910784,   7151360,  15501440,   19421472,   13003696,    3587227;
  8:  65536, 1261568, 10350592,  47187968, 130512640,  223882880,  232300960,  133175792,   32285041;
  9: 262144, 5701632, 53870592, 289787904, 976733184, 2136295680, 3028519680, 2680919520, 1343864820, 290565367;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (8*n-4+(46*n-31)*x+(84*n-66)*x^2+(62*n-55)*x^3+(16*n-16)*x^4)*P(n-1,x) - (4*x+28*x^2+60*x^3+52*x^4+16*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 + (46*n - 31)*x + (84*n - 66)*x^2 + (62*n - 55)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*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 + (46*n - 31)*x + (84*n - 66)*x^2 + (62*n - 55)*x^3 + (16*n - 16)*x^4)*v[i - 1] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*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

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