Real-rooted atlas

A081577 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(9 + 2\,x - 9\,n - n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(67 + 41\,x + 4\,x^{2} - 25\,n - 14\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(50 + 53\,x + 16\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2138122
35147131112
4205250106181580
581912457141902312992
6327759644311156829928115448
713107277872424311203293743333179744

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(4 + 2\,x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-fbe62a60256a6a00 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-40666f6b4adccd85)
Fingerprint
f6-487bd17ac511de6a

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:     13,      8,      1;
  3:     51,     47,     13,      1;
  4:    205,    250,    106,     18,      1;
  5:    819,   1245,    714,    190,     23,     1;
  6:   3277,   5964,   4311,   1568,    299,    28,    1;
  7:  13107,  27787,  24243,  11203,   2937,   433,   33,   1;
  8:  52429, 126878, 129688,  72746,  24346,  4946,  592,  38,  1;
  9: 209715, 570425, 668488, 441224, 179938, 46790, 7720, 776, 43, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-9*n+9-(n-2)*x)*P(n-1,x) - (25*n-67+(14*n-41)*x+(n-4)*x^2)*d/dx P(n-1,x) + (50+53*x+16*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[((3*n^2 - 9*n + 9 - (n - 2)*x)*P[n - 1, x] - (25*n - 67 + (14*n - 41)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (50 + 53*x + 16*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] = ((3*n^2 - 9*n + 9 - (n - 2)*x)*v[i - 1] - (25*n - 67 + (14*n - 41)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (50 + 53*x + 16*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