Real-rooted atlas

A110441 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-4\,x - 20\,x^{2} - 36\,x^{3} - 28\,x^{4} - 8\,x^{5} + 2\,n + 18\,n x + 50\,n x^{2} + 62\,n x^{3} + 36\,n x^{4} + 8\,n x^{5}\right)\,P_{n-1}(x) + \left(-x - 11\,x^{2} - 35\,x^{3} - 49\,x^{4} - 32\,x^{5} - 8\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2181423
31124448105
411690208164479
51201525449085602185
612423011202904377619129967
7128324200070801430415200652845465

All 15 rows

Technical details
Node id
n-5e3fe73521ba8587 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-010ab161ddea77e9

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: 1,  4;
  2: 1,  8,  14;
  3: 1, 12,  44,   48;
  4: 1, 16,  90,  208,   164;
  5: 1, 20, 152,  544,   908,   560;
  6: 1, 24, 230, 1120,  2904,  3776,   1912;
  7: 1, 28, 324, 2000,  7080, 14304,  15200,   6528;
  8: 1, 32, 434, 3248, 14620, 40384,  66608,  59776,  22288;
  9: 1, 36, 560, 4928, 26964, 94864, 213984, 297600, 230992, 76096;
%F Formula
The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (2*n+(18*n-4)*x+(50*n-20)*x^2+(62*n-36)*x^3+(36*n-28)*x^4+(8*n-8)*x^5)*P(n-1,x) - (x+11*x^2+35*x^3+49*x^4+32*x^5+8*x^6)*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[((2*n + (18*n - 4)*x + (50*n - 20)*x^2 + (62*n - 36)*x^3 + (36*n - 28)*x^4 + (8*n - 8)*x^5)*P[n - 1, x] - (x + 11*x^2 + 35*x^3 + 49*x^4 + 32*x^5 + 8*x^6)*D[P[n - 1, x], x])/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)]];
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] = ((2*n + (18*n - 4)*x + (50*n - 20)*x^2 + (62*n - 36)*x^3 + (36*n - 28)*x^4 + (8*n - 8)*x^5)*v[i - 1] - (x + 11*x^2 + 35*x^3 + 49*x^4 + 32*x^5 + 8*x^6)*deriv(v[i - 1], x))/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)); 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

Candidate equalities ?

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