Real-rooted atlas

A103327 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-4 - 26\,x - 52\,x^{2} - 42\,x^{3} - 12\,x^{4} + 8\,n + 40\,n x + 68\,n x^{2} + 48\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 24\,x^{2} - 48\,x^{3} - 40\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
2165644116
3643526083281352
4256192051845952244815760
51024972835840638725497618272183712
64096471042201605345287101444890881363842141504
71638422118412533763860480697344073820164238336101798424963200

All 15 rows

Technical details
Node id
n-f03b4dcaa53a2382 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-65206dde5faab998

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,       6;
  2:     16,      56,       44;
  3:     64,     352,      608,       328;
  4:    256,    1920,     5184,      5952,      2448;
  5:   1024,    9728,    35840,     63872,     54976,     18272;
  6:   4096,   47104,   220160,    534528,    710144,    489088,    136384;
  7:  16384,  221184,  1253376,   3860480,   6973440,   7382016,   4238336,   1017984;
  8:  65536, 1015808,  6766592,  25280512,  57896960,  83177472,  73168896,  36022272,   7598336;
  9: 262144, 4587520, 35127296, 154370048, 428818432, 780443648, 930201600, 699912192, 301618176, 56714752;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (8*n-4+(40*n-26)*x+(68*n-52)*x^2+(48*n-42)*x^3+(12*n-12)*x^4)*P(n-1,x) - (4*x+24*x^2+48*x^3+40*x^4+12*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 + (40*n - 26)*x + (68*n - 52)*x^2 + (48*n - 42)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] - (4*x + 24*x^2 + 48*x^3 + 40*x^4 + 12*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 + (40*n - 26)*x + (68*n - 52)*x^2 + (48*n - 42)*x^3 + (12*n - 12)*x^4)*v[i - 1] - (4*x + 24*x^2 + 48*x^3 + 40*x^4 + 12*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