Real-rooted atlas

A081578 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(11 + 2\,x - 12\,n - n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(87 + 51\,x + 4\,x^{2} - 33\,n - 18\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(66 + 69\,x + 20\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2159125
36357151136
4279333135211769
5121518099992492714300
6531994776615227739933124121
7232474824940743179374383585391135184

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(6 + 3\,x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-7b1a2203ac9e3633 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ab89b07ec2e2ca00)
Fingerprint
f6-e783f5437ef9937b

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:     15,       9,       1;
  3:     63,      57,      15,      1;
  4:    279,     333,     135,     21,      1;
  5:   1215,    1809,     999,    249,     27,     1;
  6:   5319,    9477,    6615,   2277,    399,    33,     1;
  7:  23247,   48249,   40743,  17937,   4383,   585,    39,    1;
  8: 101655,  240813,  238599, 128061,  40311,  7533,   807,   45,  1;
  9: 444447, 1183329, 1345815, 852633, 329103, 79569, 11943, 1065, 51, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n^2-12*n+11-(n-2)*x)*P(n-1,x) - (33*n-87+(18*n-51)*x+(n-4)*x^2)*d/dx P(n-1,x) + (66+69*x+20*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[((4*n^2 - 12*n + 11 - (n - 2)*x)*P[n - 1, x] - (33*n - 87 + (18*n - 51)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (66 + 69*x + 20*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] = ((4*n^2 - 12*n + 11 - (n - 2)*x)*v[i - 1] - (33*n - 87 + (18*n - 51)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (66 + 69*x + 20*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

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