Real-rooted atlas

A257625 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
297272153
327702194412963969
4816480375846220831104137457
524358806641520198288023328009331205949369
6729530712103284725318784011057472010077696033592320308991753
7218747807821612956241302857136426202560066428812804938071040141087744018722791089

All 15 rows

Technical details
Node id
n-b29e1efbc9cdcce1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2a2e8db5a042c827

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:     3,         6;
  2:     9,        72,          72;
  3:    27,       702,        1944,         1296;
  4:    81,      6480,       37584,        62208,         31104;
  5:   243,     58806,      641520,      1982880,       2332800,         933120;
  6:   729,    530712,    10328472,     53187840,     110574720,      100776960,       33592320;
  7:  2187,   4780782,   161295624,   1302857136,    4262025600,     6642881280,     4938071040,     1410877440;
  8:  6561,  43040160,  2476803744,  30263321088,  146343262464,   347075850240,   431728496640,   270888468480,    67722117120;
  9: 19683, 387400806, 37668538080, 680112210240, 4677587792640, 15843800931840, 29332141977600, 30322577940480, 16456474460160, 3656994324480;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*x^4)*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[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*deriv(v[i - 1], x))/(1 + 2*x + 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. [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