Real-rooted atlas

A257627 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
278789165
315602340834274761
4421208424050622850281185565
5143208035802003034200971100855062439103329
6587152801761458401929096009224280017620278856518729538491045
72818333440986416704013252468320847075320025662327603106794608569554218737291205961

All 15 rows

Technical details
Node id
n-b9fa97756c104cb4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-db62b20fb03c4b11

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:             6,              3;
  2:            78,             78,              9;
  3:          1560,           2340,            834,             27;
  4:         42120,          84240,          50622,           8502,             81;
  5:       1432080,        3580200,        3034200,         971100,          85506,            243;
  6:      58715280,      176145840,      192909600,       92242800,       17620278,         856518,           729;
  7:    2818333440,     9864167040,    13252468320,     8470753200,     2566232760,      310679460,       8569554,        2187;
  8:  155008339200,   620033356800,   987238717920,   791599404960,   331881633720,    67803175440,    5392955022,    85708662,      6561;
  9: 9610517030400, 43247326636800, 79629193013760, 76881317909760, 41364752934960, 12119016382200, 1735135311000, 92794447980, 857125986, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (7*n-1+(7*n-4)*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 - 1 + (7*n - 4)*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 - 1 + (7*n - 4)*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