Real-rooted atlas

A121574 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(18\,x^{2} + 26\,x^{3} + n - n x - 28\,n x^{2} - 39\,n x^{3} + 4\,n^{2} x + 10\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(16\,x^{2} + 64\,x^{3} + 52\,x^{4} - n x - 12\,n x^{2} - 33\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 11\,x^{3} + 23\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
2171018
3111353380
411576155109356
51191334296423601584
61232069192160255011897048
712729516895479101019841392731360

All 15 rows

Technical details
Node id
n-bd000e7539a3be60 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-98ef740d5236d6f3

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,  3;
  2: 1,  7,  10;
  3: 1, 11,  35,   33;
  4: 1, 15,  76,  155,   109;
  5: 1, 19, 133,  429,   642,   360;
  6: 1, 23, 206,  919,  2160,  2550,   1189;
  7: 1, 27, 295, 1689,  5479, 10101,   9841,   3927;
  8: 1, 31, 400, 2803, 11671, 29617,  44854,  37189,  12970;
  9: 1, 35, 521, 4325, 22064, 71798, 149285, 191693, 138305, 42837;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(4*n^2-n)*x+(10*n^2-28*n+18)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - (n*x+(12*n-16)*x^2+(33*n-64)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (x^2+11*x^3+23*x^4+13*x^5)*d^2/dx^2 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[((n + (4*n^2 - n)*x + (10*n^2 - 28*n + 18)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - (n*x + (12*n - 16)*x^2 + (33*n - 64)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (x^2 + 11*x^3 + 23*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(n)]];
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] = ((n + (4*n^2 - n)*x + (10*n^2 - 28*n + 18)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - (n*x + (12*n - 16)*x^2 + (33*n - 64)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (x^2 + 11*x^3 + 23*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(n)); 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. [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