Real-rooted atlas

A059473 → γ → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2628
316824
44432480
512011224256
63283761208832
78961216512642688
8244838402016384168704
96688119047488192016028160
10182723638426688864011203291136
114992010995292160360966400384294912
1213638432921631046414284832320307264954368
133726089781761025152542208149632197128963088384
14101798428875523329664199104064960011020880641289994240

Other recurrences

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

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-2}(x) + \left(2\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
Technical details
Node id
n-e9d5ee5f07302b44 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-b3ad603fc54c352a)
Fingerprint
f6-0d853bc775b295f0

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:    2;
  2:    6,     2;
  3:   16,     8;
  4:   44,    32,    4;
  5:  120,   112,   24;
  6:  328,   376,  120,    8;
  7:  896,  1216,  512,   64;
  8: 2448,  3840, 2016,  384,  16;
  9: 6688, 11904, 7488, 1920, 160;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n-2+(2*n-2)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx 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 + (2*n - 2)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(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 + (2*n - 2)*x)*v[i - 1] - (6 + 10*x + 4*x^2)*deriv(v[i - 1], 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

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741