Real-rooted atlas

A257612 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
23232468
338457620881176
4614412288742412801627152
512288030720026112084480774432783456
62949120884736097075204669440906752465926427126848
7825753602890137603887923202494464007625318494402562798081281095801216

All 15 rows

Technical details
Node id
n-37818242788af2ca · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2a2e8db5a042c827

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:           4,            2;
  2:          32,           32,            4;
  3:         384,          576,          208,            8;
  4:        6144,        12288,         7424,         1280,           16;
  5:      122880,       307200,       261120,        84480,         7744,           32;
  6:     2949120,      8847360,      9707520,      4669440,       906752,        46592,          64;
  7:    82575360,    289013760,    388792320,    249446400,     76253184,      9440256,      279808,       128;
  8:  2642411520,  10569646080,  16845373440,  13542359040,   5710086144,   1180827648,    96641024,   1679360,      256;
  9: 95126814720, 428070666240, 788759838720, 762996326400, 412133621760, 121674792960, 17691279360, 979845120, 10077184, 512;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+(4*n-2)*x)*P(n-1,x) - (4*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[(4*n + (4*n - 2)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x]];
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 + (4*n - 2)*x)*v[i - 1] - (4*x + 4*x^2)*deriv(v[i - 1], x)); 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