Real-rooted atlas

A257618 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 6\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
248484100
3960144049682904
4268805376031872499216117520
59676802419200203520063360049984326105696
642577920127733760139238400655872001200460849996864387641920
7221405184077491814401037740032065705472001945359360222082560499993612829083622784

All 15 rows

Technical details
Node id
n-9f974a244ae6ecb1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d6ac9abcd3bcb1be

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:            48,             48,              4;
  3:           960,           1440,            496,              8;
  4:         26880,          53760,          31872,           4992,             16;
  5:        967680,        2419200,        2035200,         633600,          49984,             32;
  6:      42577920,      127733760,      139238400,       65587200,       12004608,         499968,            64;
  7:    2214051840,     7749181440,    10377400320,     6570547200,     1945359360,      222082560,       4999936,         128;
  8:  132843110400,   531372441600,   844064686080,   672390512640,   277868666880,    55020994560,    4057485312,    49999872,       256;
  9: 9033331507200, 40649991782400, 74710965288960, 71788416860160, 38243673538560, 10988122521600, 1511695564800, 73634734080, 499999744, 512;
%F Formula
The row polynomials satisfy P(n,x) = (8*n-4+(8*n-6)*x)*P(n-1,x) - (8*x+8*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[(8*n - 4 + (8*n - 6)*x)*P[n - 1, x] - (8*x + 8*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] = (8*n - 4 + (8*n - 6)*x)*v[i - 1] - (8*x + 8*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