Real-rooted atlas

A257617 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 5\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
24444492
3792118841282400
419800396002354037401686696
56336001584000133452041778033724324003656
624710400741312008088300038214000705544430364464225297752
711366784003978374400533111040033818400001005919992116227188273305212814952883560

All 15 rows

Technical details
Node id
n-a466434b299cb969 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8c4e3f9383454e05

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:            44,             44,              4;
  3:           792,           1188,            412,              8;
  4:         19800,          39600,          23540,           3740,             16;
  5:        633600,        1584000,        1334520,         417780,          33724,            32;
  6:      24710400,       74131200,       80883000,       38214000,        7055444,        303644,           64;
  7:    1136678400,     3978374400,     5331110400,     3381840000,     1005919992,     116227188,      2733052,         128;
  8:   60243955200,   240975820800,   382961779200,   305469964800,   126603199800,   25228249200,   1889698580,    24597980,       256;
  9: 3614637312000, 16265867904000, 29906008704000, 28763646912000, 15353691091200, 4428802224000, 614264306040, 30505755060, 221382844, 512;
%F Formula
The row polynomials satisfy P(n,x) = (7*n-3+(7*n-5)*x)*P(n-1,x) - (7*x+7*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[(7*n - 3 + (7*n - 5)*x)*P[n - 1, x] - (7*x + 7*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] = (7*n - 3 + (7*n - 5)*x)*v[i - 1] - (7*x + 7*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