Real-rooted atlas

A143685 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(23 + 2\,x - 30\,n - n x + 10\,n^{2}\right)\,P_{n-1}(x) + \left(207 + 111\,x + 4\,x^{2} - 81\,n - 42\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(162 + 165\,x + 44\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22715143
3135117271280
48919993513912281
551037209359172951116684
631347527313329196391251631128323
7185895365229282123107649208711917751963760

All 15 rows

Technical details
Node id
n-549939118a9956ba · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9534464459a0c5cc

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,        1;
  2:      27,       15,        1;
  3:     135,      117,       27,       1;
  4:     891,      999,      351,      39,       1;
  5:    5103,     7209,     3591,     729,      51,      1;
  6:   31347,    52731,    33291,    9639,    1251,     63,     1;
  7:  185895,   365229,   282123,  107649,   20871,   1917,    75,    1;
  8: 1121931,  2512863,  2285415, 1078191,  279531,  39015,  2727,   87,  1;
  9: 6711903, 16907697, 17734383, 9996777, 3261303, 618921, 65799, 3681, 99, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (10*n^2-30*n+23-(n-2)*x)*P(n-1,x) - (81*n-207+(42*n-111)*x+(n-4)*x^2)*d/dx P(n-1,x) + (162+165*x+44*x^2+x^3)*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[((10*n^2 - 30*n + 23 - (n - 2)*x)*P[n - 1, x] - (81*n - 207 + (42*n - 111)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (162 + 165*x + 44*x^2 + x^3)*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] = ((10*n^2 - 30*n + 23 - (n - 2)*x)*v[i - 1] - (81*n - 207 + (42*n - 111)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (162 + 165*x + 44*x^2 + x^3)*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 ?

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