Real-rooted atlas

A375853 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3} + n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
281624
3209696212
4403367685121656
57089634565120256012102
611220161152028160307201228884816
716840323168011264019968017203257344577576
824073927603236608093184012902409175042621443851472

All 15 rows

Technical details
Node id
n-dbf55b6b36baa8f7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4748f217d058d591

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   2;
   2:   8,    16;
   3:  20,    96,     96;
   4:  40,   336,    768,     512;
   5:  70,   896,   3456,    5120,     2560;
   6: 112,  2016,  11520,   28160,    30720,    12288;
   7: 168,  4032,  31680,  112640,   199680,   172032,    57344;
   8: 240,  7392,  76032,  366080,   931840,  1290240,   917504,   262144;
   9: 330, 12672, 164736, 1025024,  3494400,  6881280,  7798784,  4718592,  1179648;
  10: 440, 20592, 329472, 2562560, 11182080, 29245440, 46792704, 44826624, 23592960, 5242880;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n+2+(4*n+8)*x+(5*n+10)*x^2+(2*n+4)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (4*n + 8)*x + (5*n + 10)*x^2 + (2*n + 4)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (4*n + 8)*x + (5*n + 10)*x^2 + (2*n + 4)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (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, v[i])))};
T(10)

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