Real-rooted atlas

A166343 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 14 + 14\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11414129
25684301171
32805603426211245
41680420036081212126110827
51176035280390401928040142541109629
694080329280445920291600931761280451011267371
78467203386880542640044251201923880423920399421022116473885
884672003810240070506240689606403809400011739000185544812289220461237849867

All 15 rows

Technical details
Node id
n-5de7394bf8d53953 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-addc5ba4413954b4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         14,         14,           1;
   2:         56,         84,          30,           1;
   3:        280,        560,         342,          62,           1;
   4:       1680,       4200,        3608,        1212,         126,          1;
   5:      11760,      35280,       39040,       19280,        4014,        254,          1;
   6:      94080,     329280,      445920,      291600,       93176,      12804,        510,         1;
   7:     846720,    3386880,     5426400,     4425120,     1923880,     423920,      39942,      1022,        1;
   8:    8467200,   38102400,    70506240,    68960640,    38094000,   11739000,    1855448,    122892,     2046,       1;
   9:   93139200,  465696000,   977477760,  1115735040,   749382480,  298998000,   67972240,   7913360,   374814,    4094,    1;
  10: 1117670400, 6147187200, 14431737600, 18838915200, 14920940160, 7338663360, 2201821440, 379428000, 33152696, 1136724, 8190, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 14 + 14*x + x^2.
%t Mathematica
P[1, x_] := 14 + 14*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 2 + (n + 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 14 + 14*x + x^2; for(n = 2, N, my(i = n); v[i] = (n + 2 + (n + 1)*x)*v[i - 1] - (x + 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, v[i])))};
T(10)

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