Real-rooted atlas

A119808 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 6\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 21\,n + 4\,n x + 25\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 29\,x - 8\,x^{2} + 9\,n - 19\,n x - 24\,n x^{2}\right)\,P_{n-1}'(x) + \left(-9\,x - 8\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1314
29514
32721250
4818116178
5243297904634
67291053432442258
721873645189031288042

All 15 rows

Technical details
Node id
n-a61451e00b31df2c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e8cd0fa6633bf52f

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,     1;
  2:     9,     5;
  3:    27,    21,     2;
  4:    81,    81,    16;
  5:   243,   297,    90,    4;
  6:   729,  1053,   432,   44;
  7:  2187,  3645,  1890,  312,   8;
  8:  6561, 12393,  7776, 1800, 112;
  9: 19683, 41553, 30618, 9180, 960, 16;
%F Formula
The row polynomials satisfy (9*n^2+6*n+1)*P(n,x) = (25*n^2+21*n+2+(12*n^2+4*n)*x)*P(n-1,x) + (9*n-9-(19*n+29)*x-(24*n+8)*x^2)*d/dx P(n-1,x) - (9*x+8*x^2)*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[((25*n^2 + 21*n + 2 + (12*n^2 + 4*n)*x)*P[n - 1, x] + (9*n - 9 - (19*n + 29)*x - (24*n + 8)*x^2)*D[P[n - 1, x], x] - (9*x + 8*x^2)*D[P[n - 1, x], {x, 2}])/(9*n^2 + 6*n + 1)]];
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] = ((25*n^2 + 21*n + 2 + (12*n^2 + 4*n)*x)*v[i - 1] + (9*n - 9 - (19*n + 29)*x - (24*n + 8)*x^2)*deriv(v[i - 1], x) - (9*x + 8*x^2)*deriv(deriv(v[i - 1], x), x))/(9*n^2 + 6*n + 1)); 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